Question

Find the number of terms and the sum of the terms in the AP: 3, 8, 13, 18, ..., 78.

• A. 16
• B. Number of terms =16 Sum = 652
• C. Number of terms =15 Sum = 648
• D. Number of terms =16 Sum = 695

Answer 1

The correct option is A 16

The formula to find the $n^{th}$ term of an arithmetic progression is $t_n=a+(n-1)d$.

where,

'$a$' is the first term,

'd' is the common difference,

'n' is the number of terms,

'$t_n$' is the $n^{th}$ term.

Given, $t_n$ = 78, $a_1$=3, $a_2$ = 8

$d=8-3=5$
$t_n=78$
Substituting the values in the formula, we get
$t_n=3+(n-1)5$
$\Rightarrow 78=3+(n-1)5$
$\Rightarrow 75=5n-5$
$\Rightarrow 80=5n$
$\Rightarrow n=\frac{80}{5}=16$
$\Rightarrow$ Number of terms in the given AP is 16.

Sum of terms in an A.P =
$S_n=\frac{n}{2}[2a_1+(n-1\left)d\right]$
where $S_n$ is the sum of n terms of the AP,

'n' is the number of terms of the AP,

'$a_1$' is the first term of the AP

'd' is the common difference.

Given, $a_1$ = 3, $a_2$ = 8

d = $a_2$ - $a_1$ = 8-3 = 5
$S_n$ = $\left(\frac{16}{2}\right)\left[2\right(3)+(n-1\left)\right(5\left)\right]$

$S_n$ = $\left(8\right)[6+(15)\times 5]$
$\Rightarrow$$S_n$ = $\left(8\right)[6+75]$ = $8\times 81$ = 648