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

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 = ______________

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

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

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

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

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

Compute 102⁴

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

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

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

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

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.

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

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

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

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

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.

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

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 \(P^{2}=(\frac{S}{R})^{n}\)