Question

The sum of the series $1+\left( 1+2 \right)+\left( 1+2+3 \right)+.........$ up to $n$ terms will be:
(1) ${{n}^{2}}-2n+6$
(2) $\dfrac{n\left( n+1 \right)\left( 2n-1 \right)}{6}$
(3) ${{n}^{2}}+2n+6$
(4) $\dfrac{n\left( n+1 \right)\left( n+2 \right)}{6}$

Answer 1

Here in this question we have been asked to find the sum of the series $1+\left( 1+2 \right)+\left( 1+2+3 \right)+.........$ up to $n$ terms. We know that the sum of the first $n$ positive natural numbers will be given as $\dfrac{n\left( n+1 \right)}{2}$ . This will be the general term here.

Complete step by step solution:
Now considering from the question we have been asked to find the sum of the series $1+\left( 1+2 \right)+\left( 1+2+3 \right)+.........$ up to $n$ terms.
From the basic concepts, we know that the sum of the first $n$ positive natural numbers will be given as $\dfrac{n\left( n+1 \right)}{2}$ .
Hence we can say that the general term of the series is given as $\dfrac{n\left( n+1 \right)}{2}$ .
Hence the sum of the given series will be given as $\sum\limits_{r=1}^{n}{\dfrac{r\left( r+1 \right)}{2}}$ .
By simplifying this further we will have $\Rightarrow \dfrac{1}{2}\sum\limits_{r=1}^{n}{{{r}^{2}}}+\dfrac{1}{2}\sum\limits_{r=1}^{n}{r}$ .
We know that $\sum\limits_{r=1}^{n}{r}=\dfrac{n\left( n+1 \right)}{2}$ and $\sum\limits_{r=1}^{n}{{{r}^{2}}}=\dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6}$ . By using these formulae in the expression we have we will get $\Rightarrow \dfrac{1}{2}\left( \dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6} \right)+\dfrac{1}{2}\left( \dfrac{n\left( n+1 \right)}{2} \right)$ .
By simplifying this further we will have
$\begin{aligned} &= \left( \dfrac{n\left( n+1 \right)}{4} \right)\left( \dfrac{2n+1}{3}+1 \right) \\\ &= \left( \dfrac{n\left( n+1 \right)}{4} \right)\left( \dfrac{2n+4}{3} \right) \\\ &= \dfrac{n\left( n+1 \right)\left( n+2 \right)}{6} \\\ \end{aligned}$ .
Therefore we can conclude that the sum of the series $1+\left( 1+2 \right)+\left( 1+2+3 \right)+.........$ up to $n$ terms will be given as $\dfrac{n\left( n+1 \right)\left( n+2 \right)}{6}$ .
Hence we will mark the option “4” as correct.

Note: In the process of answering questions of this type, we generally get tense by seeing the series directly and lose our confidence and make mistakes due to pressure. This is a very simple question and it can be answered easily. The first thing is to deduct the general term of any given series for answering this type of question.