The sum of a G.P. with common ratio 3 is 364, | vedclass

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(A) Let the first term be $a$, the number of terms be $n$, and the common ratio be $r = 3$.The sum of a $G.P.$ is given by $S_n = \frac{a(r^n - 1)}{r

Vedclass · Accepted Answer

$6$

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(A) Let the first term be $a$, the number of terms be $n$, and the common ratio be $r = 3$.The sum of a $G.P.$ is given by $S_n = \frac{a(r^n - 1)}{r - 1} = 364$.The last term $l$ is given by $l = ar^{n-1} = 243$.We can rewrite the sum formula as $S_n = \frac{ar^{n-1} \cdot r - a}{r - 1} = 364$.Substituting $ar^{n-1} = 243$ and $r = 3$ into the equation:$\frac{243 \cdot 3 - a}{3 - 1} = 364$$\frac{729 - a}{2} = 364$$729 - a = 728$$a = 1$.Now, using the last term formula $ar^{n-1} = 243$:$1 \cdot 3^{n-1} = 243$$3^{n-1} = 3^5$$n - 1 = 5$$n = 6$.Thus, the number of terms is $6$.

The sum of a $G.P.$ with common ratio $3$ is $364$, and the last term is $243$. Find the number of terms.

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