Question

Find the general term and sum of n terms of the series $1+6+24+84+...$

Answer 1

To find the general term and sum of n terms of the series

$1+6+24+84+......$

The scale of relation of the series $1+6x+24x^2+84x^3+....$

is $1-5x+6x^2$ and the generating function is

$=\frac{1+x}{1-5x+6x^2}$

This Expression is Equivalent to the partial fractions

$=\frac{4}{1-3x}-\frac{3}{1-2x}$

If these expressions be expanded in ascending powers of x the general term is $({4.3}^r-{3.2}^r)x^r$ $r\in \{0,1,2,3,\mathrm{4.....}\}$

Hence the general term of the given series is $({4.3}^r-{3.2}^r)$; and

The sum of n terms is $2(3^n-1)-3(2^n-1)$