Question

Find the sum of the first $30$ terms of an A.P. whose $n^{th}$ term is $3+2n$

Answer 1

It is given that the $n$th term of A.P is $T_n=3+2n$.

We know that the general term of an arithmetic progression with first term $a$ and common difference $d$ is $T_n=a+(n-1)d$, therefore,

$T_n=3+2n=5+(n-1)2$

Comparing the above term by the general term of A.P, we get $a=5$ and $d=2$

We also know that the sum of an arithmetic series with first term $a$ and common difference $d$ is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Now to find the sum of first $30$ terms of an A.P, substitute $n=30,a=5$ and $d=2$ in $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$ as follows:

$S_{30}=\frac{30}{2}\left[\right(2\times 5)+(30-1\left)2\right]=15[10+(29\times 2\left)\right]=15(10+58)=15\times 68=1020$

Hence, the sum of first $30$ terms is $1020$.