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Q.
If first term of a G.P. is 729 and its seventh term is 64 , then sum of first seven terms is
Sequences and Series
Solution:
Given, $a=729, T_7=64$
$ \Rightarrow a r^{7-1}=64 \left(\because T_n=a r^{n-1}\right) $
$ \Rightarrow 729 r^6=64$
$ \Rightarrow r^6=\frac{64}{729}=\left(\frac{2}{3}\right)^6$
On comparing base of the power 6 on both sides, we get
$\Rightarrow r=\frac{2}{3} < 1$
Now, $ S_n=\frac{a\left(1-r^n\right)}{1-r} $
$(\because r < 1)$
$\therefore S_7=\frac{729\left[1-\left(\frac{2}{3}\right)^7\right]}{1-\frac{2}{3}}=\frac{729\left[1-\frac{2^7}{3^7}\right]}{\frac{1}{1}-\frac{2}{3}}$
$=\frac{729\left[\frac{1}{1}-\frac{128}{2187}\right]}{\frac{3-2}{3}}$
$=\frac{729 \times 3}{1} \times \frac{2187-128}{2187}$
$=\frac{1}{1} \times 2059$
$=2059$