Find the sum of the geometric series 3 + 6 + 12 + … + 1536.
Tn represents the $n^{\text {th }}$ term of a G.P. series.
$r=6 \div 3=2$
$\mathrm{T}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$
$\Rightarrow 1536=3 \times 2^{n-1}$
$\Rightarrow 1536 \div 3=2^{n} \div 2$
$\Rightarrow 1536 \div 3 \times 2=2^{n}$
$\Rightarrow 1024=2^{n}$
$\Rightarrow 2^{10}=2^{n}$
∴ n = 10
Sum of a G.P. series is represented by the formula, $S_{n}=a \frac{r^{n}-1}{r-1}$ hen r>1. ‘Sn’ represents the sum of the G.P. series upto nth terms, ‘a’ represents the first term, ‘r’ represents the common ratio and ‘n’ represents the number of terms.
Here,
a = 3
r = 2
n = 10 terms
$\therefore \mathrm{S}_{\mathrm{n}}=3 \times \frac{2^{10}-1}{2-1}$
$\Rightarrow \mathrm{S}_{\mathrm{n}}=3 \times(1024-1)$
$\Rightarrow \mathrm{S}_{\mathrm{n}}=3 \times 1023$
$\therefore \mathrm{S}_{\mathrm{n}}=3069$
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