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

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

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

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

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

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

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

Compute 9⁷

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

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.

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

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

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

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

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

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