# If 1 + 1 + 2 2 + 1 + 2 + 3 3 + . . . . to N Terms is S, Then S is Equal to - Mathematics

MCQ

If $1 + \frac{1 + 2}{2} + \frac{1 + 2 + 3}{3} + . . . .$ to n terms is S, then S is equal to

#### Options

• $\frac{n (n + 3)}{4}$

• $\frac{n (n + 2)}{4}$

• $\frac{n (n + 1) (n + 2)}{6}$

•  n2

#### Solution

$\frac{n (n + 3)}{4}$

Let $T_n$ be the nth term of the given series.
Thus, we have:

$T_n = \frac{1 + 2 + 3 + 4 + 5 + . . . + n}{n} = \frac{n\left( n + 1 \right)}{2n} = \frac{n}{2} + \frac{1}{2}$

Now, let

$S_n$  be the sum of n terms of the given series.
Thus, we have:

$S_n = \sum^n_{k = 1} \left( \frac{k}{2} + \frac{1}{2} \right)$

$\Rightarrow S_n = \sum^n_{k = 1} \frac{k}{2} + \frac{n}{2}$

$\Rightarrow S_n = \frac{n\left( n + 1 \right)}{4} + \frac{n}{2}$

$\Rightarrow S_n = \frac{n}{2}\left( \frac{n + 1}{2} + 1 \right)$

$\Rightarrow S_n = \frac{n}{2}\left( \frac{n + 3}{2} \right)$

$\Rightarrow S_n = \frac{n\left( n + 3 \right)}{4}$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 21 Some special series
Q 6 | Page 20