If a, 8, b are in AP, a, 4, b are in GP, and if a, x, b are in HP then x is

The sequence \(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } }, \frac { 1 }{ \sqrt { 3 } +2\sqrt { 2 } },...... \)form an 

The HM of two positive numbers whose AM and GM are 16, 8 respectively is

If S_n denotes the sum of n terms of an AP whose common difference is d, the value of S_n - 2S_n-1 + S_n-2 is

The remainder when 38¹⁵ is divided by 13 is

The n^th term of the sequence 1, 2, 4, 7, 11,... is

The sum up to n terms of the series \(\frac { 1 }{ \sqrt { 1 } +\sqrt { 3 } } +\frac { 1 }{ \sqrt { 3 } +\sqrt { 5 } } +\frac { 1 }{ \sqrt { 5 } +\sqrt { 7 } } +\)....is 

The n^th term of the sequence \(\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 8 } ,\frac { 15 }{ 6 } \),......is

The sum up to n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\).....is

The value of the series\(\frac { 1 }{ 2 } +\frac { 7 }{ 4 } +\frac { 13 }{ 8 } +\frac { 19 }{ 16 } +\).....is

The sum of an infinite GP is 18. If the first term is 6, the common ratio is

The coefficient of x⁵ in the series e^-2x is

The value of \(\frac { 1 }{ 2! } +\frac { 1 }{ 4! } +\frac { 1 }{ 6! } +....is\)

The value of \(1-\frac { 1 }{ 2 } \left( \frac { 2 }{ 3 } \right) +\frac { 1 }{ 3 } { \left( \frac { 2 }{ 3 } \right) }^{ 2 }-\frac { 1 }{ 4 } { \left( \frac { 2 }{ 3 } \right) }^{ 2 }+....is\)

If \(\frac { { T }_{ 2 } }{ { T }_{ 3 } } \)is the expansion of (a+b)ⁿ and \(\frac { { T }_{ 3 } }{ { T }_{ 4 } } \) is the expansion of (a+b)ⁿ⁺³ are equal, then n = ______________

The Co-efficient of x^-17 in \({ \left( { x }^{ 4 }-\frac { 1 }{ { x }^{ 3 } } \right) }^{ 15 }\)is _____________ 

If the sum of n terms of an A. P. be 3n² - n and its common difference is 6, then its first term is ______________

The term without x in \({ \left( 2x-\frac { 1 }{ 2{ x }^{ 2 } } \right) }^{ 12 }\) is ______________

The first and last term of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be ______________

If the first, second and last term of an A.P. are a, b and 2a respectively, then its sum is ______________

If in an infinite G. P. first term is equal to 10 times the sum of all successive terms, then its common ratio is ______________

The nth term of a G.P is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is ______________

The value of \({ 9 }^{ \frac { 1 }{ 3 } }\) ,\({ 9 }^{ \frac { 1 }{ 9 } }\)\({ 9 }^{ \frac { 1 }{ 27}}\),\(\infty \) is ______________

The sum of the series \(\frac { 1 }{ { log }_{ 2 }^{ 4 } } +\frac { 1 }{ { log }_{ 4 }^{ 4 } } +\frac { 1 }{ { log }_{ 8 }^{ 4 } } +.....+\frac { 1 }{ { log }_{ { 2 }^{ n } }^{ 4 } } is\)______________

If \(\Sigma n=210\) then \(\Sigma { n }^{ 2 }\)= ______________

Sum of n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } ..is\) ______________

The series 1+4x+8x² + \(\frac { 32 }{ 3 } { x }^{ 3 }+.....+\infty \ is\) ______________

The series for log \(\left( \frac { 1+x }{ 1-x } \right) is\) ______________

The Co-efficient of x³ in \(\sqrt { \frac { 1-x }{ 1+x } } ,\left| x \right| <1\ is\ \)______________

The value of 2 + 4 + 6 + + 2n is

The coefficient of x⁶ in (2 + 2x)¹⁰ is

The coefficient of x⁸y¹² in the expansion of (2x + 3y)²⁰ is

If ⁿC₁₀ > ⁿC_r for all possible r, then a value of n is

If a is the arithmetic mean and g is the geometric mean of two numbers, then

If (1 + x²)² (1 + x)ⁿ = a₀+ a₁x + a₂x² + ... + xⁿ⁺⁴ and if a₀, a₁, a₂ are in AP, then n is

With usual notation C₀ + C₂ + C₄ + ... is ______________

In the expansion of (2x + 3)⁵ the coefficient of x² is ______________

In the expansion of (1 +x )²² which term is the middle term ______________

AM, GM, HM denote the Arithmetic mean, Geometric mean and Harmonic mean respectively the relationship between this is ______________

In the series \(\frac{1}{1+\sqrt 2}+\frac{1}{\sqrt 2+\sqrt 3}+\frac{1}{\sqrt 3+\sqrt 4}+...\) some of first 24 number is ______________

1 - 2x + 3x² - 4x³ + ..., Ixl< 1 is ______________

\(\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+...\) is ______________

\(\sqrt \frac{1-2x}{1+2x}\) is approximately equal to ______________

Expansion of \(log(\sqrt \frac{1+x}{1-x})\) is ______________

The value of \(1-\frac{1}{2}(\frac{3}{4})+\frac{1}{3}(\frac{3}{4})^2-\frac{1}{4}(\frac{3}{4})^3+...\)is ______________

The coefficient of a⁵ in the expansion of (3a + 5b)⁵ is ______________

The coefficient of x³² in the expansion of \((x^4-\frac{1}{x^3})^{15}\)______________

The middle term in the expansion of  is \((x- \frac{2}{x})^{12}\) is ______________

The sum of the series C₀²- C₁² + C₂² .....+ (- 1)ⁿC²_n where n is an even integer is ______________

The ratio of the coefficient of x ¹⁵ to the term independent of x in \([x^2+(\frac{2}{x})]^{15}\) is ______________

3 log 2 + \(\frac{1}{4}-\frac{1}{2}(\frac{1}{4})^2+\frac{1}{3}(\frac{1}{4})^2\)+ . . . = ______________

\(\left(1+\frac{1}{\lfloor2}+\frac{1}{\lfloor4}+\frac{1}{\lfloor6}+...\right)^2-\left(1+\frac{1}{\lfloor3}+\frac{1}{\lfloor5}+\frac{1}{\lfloor7}+...\right)^2=\)______________

\(\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+. . .\infty =\) ______________

The larget coefficients in the expansion of (1 + X)²⁴ is ______________

Sum of the binomial coefficients is ______________

The last term in the expansion of (2 +\(\sqrt { 3 } \))⁸ ______________

The sum of the coefficients in the expansion of (1 - x)¹⁰ is ______________

The value of nC₀ - nC₁ + nC₂ - nC₃ ... + (-1)ⁿnC_n is ______________

The value of n for which \(\frac{a^{n+1}+b^{n+1}}{a^n+b^n}\) is the arithmetic mean of a and b is ______________

\(\frac{1}{q+r},\frac{1}{r+p},\frac{1}{p+q}\) are in A.P., then ______________

If a,b, c are in A.P, as well as in G.P then ______________

The sum of 40 terms of an A.P whose first term is 2 and common difference 4 will be ______________

2^1/4 4^1/8 8^1/16 16^1/32 . . . = ______________

If x, 2x + 2, 3x + 3 . . . are in G.P, then the 4^th term is ______________

If an A.P the sum of terms equidistant from the beginning and end is equal to ______________

Expand \(\left( { 2x }^{ 2 }-\frac { 3 }{ x } \right) ^{ 3 }\)

Expand \(\left( { 2x }^{ 2 }-3\sqrt { 1-{ x }^{ 2 } } \right) ^{ 4 }+({ 2x }^{ 2 }+3\sqrt { 1-{ x }^{ 2 }) } ^{ 4 }\)

Compute 102⁴

Compute 99⁴

Compute 9⁷

Using binomial theorem, indicate which of the following two number is larger (1.01)^1000000 (OR)10, 000

Find the Co-efficient of x¹⁵ in \(\left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 3 } } \right) ^{ 10 }\)

Find the Co-efficient of x⁶ and the co -efficient of x² in \(\left( { x }^{ 2 }-\frac { 1 }{ { x }^{ 3 } } \right) ^{ 6 }\)

Find the Co-efficient of x⁴ in the expansion (1+x³)⁵⁰ \(\left( { x }^{ 2 }+\frac { 1 }{ { x }^{ 3 } } \right) ^{ 5 }\)

Find the Constant term of \(\left( { 2x }^{ 3 }-\frac { 1 }{ { 3x }^{ 2 } } \right) ^{ 5 }\)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\(\frac { 1 }{ 5+x } \)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\(\frac { 2 }{ { \left( 3+4x \right) }^{ 2 } } \)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\({ \left( 5+{ x }^{ 2 } \right) }^{ \frac { 2 }{ 3 } }\)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\({ \left( x+2 \right) }^{ -\frac { 2 }{ 3 } }\)

Find \(\sqrt [ 3 ]{ 1001 } \) approximately. (two decimal places).

Prove that \(\sqrt [ 3 ]{ { x }^{ 3 }+6 } -\sqrt [ 3 ]{ { x }^{ 3 }+3 } \) is approximately equal to \(\frac { 1 }{ { x }^{ 2 } } \) when x is sufficiently large.

Prove that \(\sqrt { \frac { 1-x }{ 1+x } } \) is approximately equal to 1 - x + \(\frac{x^2}{2}\) when x is very small.

Find the last two digits of the number 3⁶⁰⁰

If n is a postive integer, show that 9ⁿ⁺¹ - 8n - 9 is always divisible by 64

if n is an odd positive integer, prove that the Co-efficients of the middle terms in the expansion equal

if the binomial co-efficients of three consecutive terms in the expansion of ( a + xⁿ) are in the radio 1:7:42 then find n

In the binomial coefficient of (1+x)ⁿ  the Coefficients of the 5^th, 6^th and 7^th terms are in A.P find all values of n

Prove that \({ C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+{ C }_{ 2 }^{ 2 }+...=\frac { (2n)! }{ (n)! } \)

Write the first 6 terms of the exponential series e^5x

Write the first 6 terms of the exponential series e^-2x

Write the first 6 terms of the exponential series \({ e }^{ \frac { 1 }{ 2 } x }\)

 If n is a positive integer and R is a nonnegative integer. prove that the co-efficients of x^r and x^n-r Expansion of (1+x)ⁿ are equal

If a and b are distinct integers, prove that aⁿ - bⁿ, is the factor of aⁿ -bⁿ, whenever n is a positive integer [Hint: write aⁿ = (a-b+b)]ⁿ

In the binomial expansion of (a+b)ⁿ the coefficients of the 4^th and 13^th terms are equal to each other, find n.

Write the first 4 terms of the logarithmic series of log (1 + 4x). Find the intervals on which the expansions are valid

Write the first 4 terms of the logarithmic series of log (1 - 2x). Find the intervals on which the expansions are valid

Write the first 4 terms of the logarithmic series of \(\log { \left( \frac { 1+3x }{ 1-3x } \right) } \). Find the intervals on which the expansions are valid

Write the first 4 terms of the logarithmic series of \(\log { \left( \frac { 1-2x }{ 1+2x } \right) } \). Find the intervals on which the expansions are valid

If \(y=x+\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 4 } }{ 4 } ....\) then show that \(x=y-\frac { { y }^{ 2 } }{ 2! } +\frac { { y }^{ 3 } }{ 3! } +\frac { { y }^{ 4 } }{ 4! } +.....\)

If p - q is small compared to either p or q, then show that \(n\sqrt { \frac { p }{ q } } =\frac { \left( n+1 \right) p+\left( n-1 \right) q }{ \left( n-1 \right) p+\left( n+1 \right) q } \)
Hence find \(8\sqrt { \frac { 15 }{ 16 } } \)

Find the coefficient of x⁴ in the expansion of \(\frac { 3-4x+{ x }^{ 2 } }{ { e }^{ 2x } } \)

Find the value of \(\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { 2 }^{ n-1 } } \left( \frac { 1 }{ { 9 }^{ n-1 } } +\frac { 1 }{ { 9 }^{ 2n-1 } } \right) } \)

Find the sum of first n terms of the series 1² + 3² + 5²+...

Find a negative value of m if the Co-efficient of x² in the expansion of (1+x)m, |x|<1 is 6

Show that \(n!>\left( \frac { n }{ e } \right) ^{ 2 }\) for n ∈ N

Find the general term in the expansion of \({ \left( \frac { 4x }{ 5 } -\frac { 5 }{ 2x } \right) }^{ 9 }\)

Find the middle term in \({ \left( x-\frac { 1 }{ 2y } \right) }^{ 10 }\)

Find the greatest term in (1 + 2x)⁸ when x = 2.

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

Write down the series whose r^th  term is \(\frac{1}{3}r\). Is it an arithmetic series?

Determine the number of terms in the G. P {T_n} if T₁ = 3, T_n = 96 and S_n = 189.

If \(x=a+\frac { a }{ r } +\frac { a }{ { r }^{ 2 } } +...+\infty ,y=b-\frac { b }{ r } +\frac { b }{ { r }^{ 2 } } +.....+\infty \quad z=c+\frac { c }{ { r }^{ 2 } } +\frac { c }{ { r }^{ 4 } } +...+\infty\) then show that \(\frac{xy}{z}=\frac{ab}{c}\)

If G is the G. M. between a and b show that \(\frac { 1 }{ { G }^{ 2 }-{ a }^{ 2 } } +\frac { 1 }{ { G }^{ 2 }-{ b }^{ 2 } } =\frac { 1 }{ { G }^{ 2 } } \)

If a, b, c are in A.P., show that (a-c)² = 4(b² - ac).

If three distinct real numbers a, b, c are in G.P and a + b + c = bx then show that \(x\le -1\quad or\quad x\ge 3\)

If H be the H. M. between a and b, then show that (H - 2a) (H - 2b) = H²

Find the n^th term of the series 3 - 6 + 9 -12 + ...

Find the expansion of (2x + 3)⁵.

Evaluate 98⁴ .

Find the middle term in the expansion of (x +y)⁶.

Find the middle terms in the expansion of (x + y)⁷.

Find the coefficient of x⁶ in the expansion of (3 + 2x)¹⁰.

Find the coefficient of x³ in the expansion of (2 - 3x)⁷.

Find the last two digits of the number 7⁴⁰⁰.

Prove that if a, b, c are in HP, if and only if \({a \over c}={a-b\over b-c}.\)

If the 5^th and 9^th terms of a harmonic progression are \({1\over 19}\) and \({1 \over 35},\) find the 12^th term of the sequence.

Find seven numbers A₁, A₂, ... , A₇ so that the sequence 4, A₁, A₂, ... , A₇, 7 is in arithmetic progression and also 4 numbers G₁, G₂, G₃, G₄ so that the sequence 12, G₁, G₂, G₃, G₄, is in geometric progression.

If the product of the 4^th, 5^th and 6^th terms of a geometric progression is 4096 and if the product of the 5^th, 6^th and 7^th terms of it is 32768, find the sum of first 8 terms of the geometric progression.

Find the term independent of x in the expansion of \((x^2+\frac{3}{x})^{15}\).

Find the coefficient x⁹ in the expansion of \((ax^2-\frac{b}{cx})^{12}\).

With usual notation find the sum C₀ + 3C₁ + 5C₂ + ... + (2n + 1)C_n where C_r is representing ⁿC_r.

Which two consecutive terms in the expansion (1 +x)¹⁵ have equal coefficients.

Find the sum up to n terms of the series : \(1+{6\over 7}+{11\over 49}+{16\over 343}+...\)

Find the sum of the first n terms of the series \({1\over 1+\sqrt{2}}+{1\over\sqrt{2}+\sqrt{3}}+{1\over\sqrt{3}+\sqrt{4}}+...\)

Find \(\sum_{k=1}^{n}{1\over k(k+1)}.\)

Find a positive value of m for which the coefficient of x² in the expansion of (1 + x)^m is 6.

In the binomial expansion of (1+a)^m+n, Prove  that the coefficients of a^m and aⁿ are equal.

Find the \(\sqrt [ 3 ]{ 126 } \) approximately to two decimal places.

If the roots of the equation (q - r) x² + (r - p)x + p - q = 0 are equal, then show that p, q and r are in A.P.

If a, b, c are respectively the p^th q^th and r^th terms of a GP. show that (q - r) log a + (r - p) log b + (p - q) log c = 0.

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetic-geometric progression, harmonic progression and none of them. \(\frac { 1 }{ 2^{ n+1 } } \)

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetic geometric progression, harmonic progression and none of them \(\frac { \left( n+1 \right) \left( n+2 \right) }{ \left( n+3 \right) (n+4) } \)

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression,arithmetic-geometric progression, harmonic progression and none of them 4\(\left( \frac { 1 }{ 2 } \right) ^{ n }\)

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetic -geometric progression, harmonic progression and none of them \(\frac { (-1)^{ n } }{ n } \)

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetic -geometric progression, harmonic progression and none of them \(\frac { 2n+3 }{ 3n+4 } \)

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetic -geometric progression, harmonic progression and none of them 2018

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetic - geometric progression, harmonic progression and none of them \(\frac { 3n-2 }{ 3^{n-1} } \)

Write the first 6 terms of the sequences whose n^th term a_n given below
\({ a }_{ n }=\begin{cases} n+1\quad if\quad n\quad is\quad odd \\ n\quad \quad if\quad n\quad is\quad even \end{cases}\)

Write the first 6 terms of the sequences whose n^th term a_n given below \(a_n= \begin{cases}n & \text { if } n \text { is } 1,2 \text { or } 3 \\ a_{n-1}+a_{n-2}+a_{n-3} & \text { if } n>3\end{cases}\)

Write the n^th term of the following sequences
2,2,4,4,6,6

Write the n^th term of the following sequences
\(\frac { 1 }{ 2 } ,\frac { 2 }{ 3 } ,\frac { 3 }{ 4 } ,\frac { 4 }{ 5 } ,\frac { 5 }{ 6 } \)

Write the n^th term of the following sequences
\(\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 5 }{ 6 } ,\frac { 7 }{ 8 } ,\frac { 9 }{ 10 } \)

Write the n^th term of the following sequences
6,10, 4, 12, 2, 14, 0, 16, -2...

The product of three increasing numbers in GP is 5832. if we add 6 to the second number and 9 to the third number, then resulting number form an AP. Find the numbers in GP

Write nth term of the Sequence \(\frac { 3 }{ { 1 }^{ 2 }{ 2 }^{ 2 } } ,\frac { 5 }{ { 2 }^{ 2 }{ 3 }^{ 2 } } ,\frac { 7 }{ { 3 }^{ 2 }{ 4 }^{ 2 } } \) as a difference of two terms 

If t_k the  kth term of a GP, then show that t_n-k, t_n, t_n+k also form a GP for any postive integer K

The first three terms in the expansion of (1 + ax)ⁿ are 1 + 12x + 64x². Find n and a

Prove that in the expansion of (1+x)ⁿ, the Co-efficient of terms equidistant from the beginning and from the end are equal

If a,b,c are in geometric progressions and if \({ a }^{ \frac { 1 }{ x } }={ b }^{ \frac { 1 }{ y } }={ c }^{ \frac { 1 }{ z } }\) , then prove that x, y, z are in arithmetic progression

Show that the sequence where log a,\(log\frac { { a }^{ 2 } }{ b^{ 1 } } log\frac { { a }^{ 2 } }{ { b }^{ 2 } } \)  ..is an A.P

The AM of two numbers exceeds their GM by 10 and HM by 16, Find the numbers

For what value of n, the nth term of the series "3 + 10 + 17 +..+ and 63 + 65 + 67 +... are equal

Find the A.P in which the sum of any number of terms is always there times the square of the number of these terms

Find the co-efficient of xⁿ  in the series 1 + (a+bx) + \(\frac { (a+bx)^2}{ 2! } +\frac { (a+bx)^{ 3 } }{ 3! } \)

Sum the series \(\frac { 2 }{ 5 } +\frac { 2 }{ { 3.5 }^{ 3 } } +\frac { 2 }{ { 5.5 }^{ 5 } } ....\infty \)

The sum of two members is\(\frac { 13 }{ 6 } \). An even number A.M.S are being inserted between them and their sum exceeds their number by 1. Find the number of A.M.S inserted.

The first term of a G.P is 1. The sum of third and fifth terms is 90. Find the common ration of the G.P

Find all the sequence which are simultaneously arithmetic and geometric progression.

If the m^th term of a H.P is n and n^th term is m, then show that its p^th  term is \(\frac{mn}{p}\).

If two G.M's g₁, g ₂ and g₃ and one A.M be inserted between two numbers then show that 2A = \(\frac { { g }_{ 1 }^{ 2 } }{ { g }_{ 2 } } +\frac { { g }_{ 2 }^{ 2 } }{ { g }_{ 1 } } \)

How many terms of the G.P.\(\sqrt { 3 } ,3\sqrt [ 3 ]{ 3 } \) Sum upto 39 + 13\(\sqrt { 13 } \) ?

The Sum of infinite number of terms of G.P is 23 and the sum of their sequence is 69. Find the G.P

If a, b, c are in A.P b, c, d are in G.P, c, d, e are in H.P then show that a, c, e in G.P

The 2^nd, 3^rd and 4^th terms in the binomial expansion of (x + a)ⁿ are 240, 720 and 1080 for a suitable value of x. Find x, a and n.

Using Binomial theorem, prove that 6ⁿ - 5n always leaves remainder 1 when divided by 25 for all positive integer n.

Expand \({\left( 2x-{1\over 2x} \right)}^{4}.\)

If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of (x+ y)ⁿ are equal.

Find the sum : \(1+{4\over5}+{7\over 25}+{10\over125}+.....\)

Insert 5 arithmetic means between 3 and 15.

If p^th term of an AP is q and q^th term is p, find (p + q)^th term.

Find 3 numbers in AP where sum is 15 and sum of their reciprocals is \(\frac{71}{105}.\)

Find 3 numbers in GP where sum is 24 and product is 216.

Find \(\sum_{1}^{\infty}{\frac{1}{(k+1)(k+2)}}\).

Expand (1+ x)^{\(2\over 3\)} up to four terms for |x| < 1.

Write the first 6 terms of the sequences whose n^th term an is given below:
\(a_n=\begin{cases} 1 \\ 2 \\{a}_{n-1}+{a}_{n-2} \\\end{cases}\)\(if\ n=1\\if\ n=2,\\if\ n>3\)

Find the coefficient of the term involving x³² and x^-17 in the expansion of \((x^{4}-\frac{1}{x^{3}})^{15}\).

If the sum of the coefficients in the expansion of (x+y)ⁿ is 4096. Then find the greatest coefficient in the expansion.

Find the coefficient of \(\frac{1}{x^{17}}\) in the expansion of \((x^{4}-\frac{1}{x^{3}})^{15}\).

If p is a real number and if the middle term in the expansion of \((\frac{p}{2}+2)^{8}\) is 1120, find p.

Find the first five terms of the sequence given by a_n =  {\({ a }_{ 1 }=1\\ { a }_{ n }={ a }_{ n-1 }+2,\quad n\ge 2\)

Find the 18^th and 25^th terms of the sequence defined by
a_n = {\(n(n+2),\quad if\quad n\quad is\quad even\quad natural\quad number\\ \frac { 4n }{ { n }^{ 2 }+1 } ,\ if\ n\ is\ odd\ natural\ number\\ \)

Write the first six terms of the sequences given by a₁ = a₂ = 1, an = a_n-1+ a_n-2 (n ≥ 3)

Write the first six terms of the sequences given by a₁ = 4, a_n+1 = 2na_n.

An A.P. consists of 21 terms. The sum of the three terms in the middle is 129 and of the last three is 237. Find the series.

Prove that the product of the 2^nd and 3^rd terms of an arithmetic progression exceeds the product of the first and fourth by twice the square of the difference between the 1^st and 2^nd.

If the p^th, q^th and r^th terms of an A.P. are a, b, c respectively, prove that a (q - r) + b (r - p) + c (p - q) = 0.

If a, b, c are in A.P. and p is the A.M. between a and b and q is the A.M. between band c, show that b is the A.M. between p and q.

If x, y, z be respectively the p^th, q^th and r^th terins ofa G.P. show that x^q-r, y^r-p, z^p-q = 1

If the ratio of the sums of m terms and n terms of an A.P. be m² : n², prove that the ratio of its mth and nth terms is (2m - 1) : (2n - 1).

The sum of first three terms of a G.P. is to the sum of the first six terms as 125: 152. Find the common ratio of the G.P.

Find the sum to n terms the series: (x + y) + (x²+ Xy + y²)+ (x²+ x²y + xy + y³) + ...

Sum the series: (1 + x) + (1 + x + x²) + (1 + x + x² +x³) + ... up to n terms

Sum up to n terms the series: 
7 + 77 + 777 + 7777 + ...

Write the first four terms in the expansions of the following 
\(\frac { 1 }{ { (2+x) }^{ 4 } } where\left| x \right| >2\)

Write the first four terms in the expansions of the following
\(\frac { 1 }{ \sqrt [ 6 ]{ 6-3x } } where\left| x \right| <2\)

Evaluate the following:
\(\sqrt [ 3 ]{ 1003 } \) correct to 4 places of decimals

Evaluate the following:
\(\frac { 1 }{ \sqrt [ 3 ]{ 128 } } \)correct to 4 places of decimals

If x so large prove that \(\sqrt { { x }^{ 2 }+25 } -\sqrt { { x }^{ 2 }+9 } =\frac { 8 }{ x } \) nearly.

Show that \({ x }^{ n }=1+n\left( 1-\frac { 1 }{ x } \right) +\frac { n(n+1) }{ 1.2 } \left( 1-\frac { 1 }{ x } \right) ^{ 2 }+...\)

Find the sum up to the 17^th term of the series \(\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +...+\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+{ 3 }^{ 3 } }{ 1+3+5 } +......\)

Compute the sum of first n terms of the following series 8 + 88 + 888 + .......

Compute the sum of first n terms of the following series 6 + 66 + 666 + .......

Compute the sum of first n terms of 1 + (1 + 4) + (1 + 4 + 4²) + (1 + 4 + 4² + 4³) + ...

Find the general terms and sum to n terms of the sequence 1, \(\frac{4}{3},\frac{7}{9},\frac{10}{27},....\)

Find the value of n if the sum to n terms of the series \(\sqrt { 3 } +\sqrt { 75 } +\sqrt { 243 } +....is\quad 435\sqrt { 3 } .\)

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

A man repays an amount of Rs. 3250 by paying Rs. 20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

In a race, 20 balls are placed in a line at intervals of 4 meters, with the first ball 24 meters away from the starting point. A contestant is required to bring the balls back to the starting place one at a time. How far would the contestant run to bring back all balls?

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

What will Rs. 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

In a certain town, a viral disease caused severe health hazards upon its people disturbing their normal life. It was found that on each day, the virus which caused the disease spread in Geometric Progression. The amount of infectious virus particle gets doubled each day, being 5 particles on the first day. Find the day when the infectious virus particles just grow over 1,50,000 units?

If P be the sum of odd terms and Q that of even terms in the expansion of (x + y)ⁿ, then prove that
(i) (x²-y²)ⁿ = P²-Q² 
(ii) (x-y)²ⁿ = 4PQ

If the Co-efficients of three successive terms in the expansion of (1 +x)ⁿ are in the ratio 1 : 3 : 5, then find the value of n

If x = 0.001, prove that \(\frac { { \left( 1-2x \right) }^{ \frac { 2 }{ 3 } }{ \left( 4+5x \right) }^{ \frac { 3 }{ 2 } } }{ \sqrt { 1-x } } \) = 8.01 up to two places of decimals 

If (p+1) ^th  term of an A.P is twice the (q+1)^th terms prove that the (3p+1)^th term is twice the  (p+q+1)^th term

A manufacture of radio sets produced 600 units the third year and 700 units in the seventh year. Assuming the production increases uniformly by a fixed number every year, find
(i) the production in the first year 
(ii) the total production in 7 years and
(iii) the production in the 10th year.

If S _n denotes that Sum of n terms of a G. P., prove that (s₁₀-s₂₀ )² = s₁₀ (s₃₀ - s₂₀)

If A and G be respectively the A. M and G. M between two positive numbers, find the numbers

If the first two terms of a H. P are \(\frac { 2 }{ 5 } \)  and\(\frac { 12 }{ 13 } \)  respectively, find the largest term of the H.P.

If the harmonic mean between two positive numbers is to their geometric mean is 12 : 13, find the ratio between the numbers.

Find the sum of the series \(1+\frac { 2 }{ 5 } +\frac { 3 }{ { 5 }^{ 2 } } +\frac { 5 }{ { 5 }^{ 3 } } +\)

Find the sum to n terms of the series 1 - 5 + 9 - 13+ ......

Find the sum of the series \(\frac { { 1 }^{ 3 } }{ 1 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 } }{ 1+3 } +\frac { { 1 }^{ 3 }+{ 2 }^{ 3 }+3^{ 3 }+.... }{ 1+3+5 } to\ n\ terms\)

Find the fourth root of 623 correct to seven places of decimal.

Find the Co-efficient of x⁵ in the expansion of \(\frac { 1-4x-{ x }^{ 2 } }{ ex } \)

If \(\alpha ,\beta \)are the roots of the equation x²-px + q = 0, then prove that \(\log { (1+px+q{ x }^{ 2 }) } =(\alpha +\beta )x=\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } { x }^{ 2 }+\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 3 } { x }^{ 3 }-....\infty \)

Expand \((x^2+\sqrt{1-x^2})^5+(x^2-\sqrt{1-x^2})^5.\)

Find \(\sum _{ n=1 }^{\infty }{1\over n^2+5n+6 } \)

Find the coefficient of x in the expansion of \(log(\frac{1}{1-5x+6x^2})\).

Find \(\sqrt{x^2+4}-\sqrt{x^2-4}\) when x is large.

Find \(\sqrt {4+x^2}-\sqrt {4-x^2}\) when x is small.

Evaluate \(\sum_{k=1}^{10}(k^2-3k+5)\).

Expand \({1\over(1+3x)^2} \) in powers of x. Find a condition on x for which the expansion is valid.

Find the sum of the first 20-terms of the arithmetic progression having the sum of first 10 terms as 52 and the sum of the first 15 terms as 77.

Expand \({1\over (3+2x)^2}\) in powers of x. Find a condition on x for which the expansion is valid.

Find \(\sqrt [ 3 ]{ 65} .\)

Prove that \(\sqrt [ 3 ]{ x^3+7 } -\sqrt [ 3 ]{ x^3+4 } \) is approximately equal to \({1\over x^2}\) when x is large.

The coefficient of (r - 1)^th, r^th, and (r + 1)^th terms in the expansion of (x + 1)ⁿ are in the ratio 1 : 3 : 5. Find both n and r.

Find the value of \((a^{2}+\sqrt{a^{2}-1})^{4}+(a^{2}-\sqrt{a^{2}-1})^{4}\)

Show that the coefficient of the middle term in the expansion of (1+x)²ⁿ is equal to the sum of the coefficients of the two middle terms in the expansion of (1+x)^2n-1.

If three consecutive coefficients in the expansion of (1+x)ⁿ are in the ratio 6:33:110,find n.

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n, terms of a G.P. , then prove that S₁ (S₃ - S₂) = (S₂ - S₁)².

If sum of the n terms of a G.P be S, their product P and the sum of their reciprocals R, then prove that \(P^{2}=(\frac{S}{R})^{n}\)