Question

Find sn of the following arithmetico geometric sequence 1, 4x, 7x^2,10x^3, 13x^4....(and so on...)

Answer 1

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Answer 2

$S_n=\frac{1+2x}{(1-x)^2}$

Step-by-step explanation:

Given sequence

$1,4x,7x^2,10x^3,13x^4,..$

$S_n=1+4x+7x^2+10x^3+13x^4+..$...(1)

Multiply by x on both sides

$xS_n=x+4x^2+7x^3+10x^4+13x^5+...$....(2)

Subtract equation (2) from (2)

$S_n-xS_n=1+3x+3x^2+3x^3+..$

$S_n(1-x)=1+3x(1+x+x^2+...)$

Sum of infinite terms in G.P

$S_n=\frac{a}{1-r}$

Where r <1

$1+x+x^2+..$ is a G.P

Because the ratio of two consecutive terms is constant

$\frac{x}{1}=\frac{x^2}{x}=x$

$r=\frac{a_2}{a_1}=\frac{x}{1}=x$

Using the formula

$S_n(1-x)=1+3x\times \frac{1}{1-x}$

$S_n(1-x)=\frac{1-x+3x}{1-x}$

$S_n=\frac{1+2x}{(1-x)^2}$

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