# Find the nth term and sum upto 13terms of the sequence: 3,−1, 1/3, -1/9....

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Find the nth term and sum up to 13terms of the sequence: 3,−1, 1/3, -1/9...

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Given sequence is :

3,-1,$\frac{1}{3}$,$\frac{-1}{9}$,....

∴ a1 = 3, a2 = -1, a3$\frac{1}{3}$, a4$\frac{-1}{9}$,....

Common ration = r = $\frac{a_4}{a_3}$ = $\frac{a_3}{a_2}$ = $\frac{a_2}{a_1}$ = $\frac{-1}{3}$ < 1

nth term of given sequence is,

an = arn-1

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

= (-1)n-1 x 3 x $\frac{1}{3^{n-1}}$

$\frac{(-1)^{n-1}}{3^{n-2}}$

Sum upto 13 terms = S13$\frac{a(1-r^{13})}{1-r}$

$\frac{3(1-(\frac{-1}{3})^{13})}{1-(\frac{-1}{3})}$

$\frac{3(1+\frac{1}{3^{13}})}{1+\frac{1}{3}}$

$\frac{3(1+\frac{1}{3^{13}})}{\frac{4}{3}}$

$\frac{9}{4}$(1 + $\frac{1}{3^{13}}$)