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 

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

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.

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

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

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