Question

If the sum of the first $2n$ terms of $2,5,8,....$ is equal to the sum of the first $n$ terms of $57,59,\mathrm{61....},$ then $n$ is equal to

• A. $10$
• B. $12$
• C. $11$
• D. $13$

Answer 1

Answer: (C)

$11$

The given series are $2,5,8,....$ to $2n$ terms and $57,59,61,....$ to $n$ terms
For first series $a=2,d=5-2=3$ and number of terms is $2n$ and for first series $a=57,d=59-57=2$ and number of terms is $n$
Formula for the sum of first terms of an AP is
$S=\frac{n}{2}[2a+(n-1)d]$ , [where $a,d,n$ and $S$ are the first term, common difference, sum of the AP and number of terms in AP]
$\frac{2n}{2}[2\times 2+(2n-1)3]=\frac{n}{2}[2\times 57+(n-1)\times 2]$
$\Rightarrow 2[4+(2n-1)3]=[114+(n-1)\times 2]$
$\Rightarrow 2[6n+1]=[2n+112]\Rightarrow 12n+2=2n+112\Rightarrow 10n=110\Rightarrow n=\frac{110}{10}\Rightarrow n=11$