Question

Find the value of $3+5+7+......\text{upto }n$ terms
A.$n\left( n-2 \right)$
B.${{\left( n+2 \right)}^{2}}$
C.$n\left( n+2 \right)$
D.${{n}^{2}}$

Answer 1

Given series in in the form of A.P. We have to use the formula of Arithmetic Progression to solve this series. These are $n$ terms in the series and the answer will be in terms of $n$ .

Formula used:
${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$
Where
$a$ is First term
$d$ is common difference
$n$ Number of terms

Complete step-by-step answer:
$3+5+7.....+$ upto $n$ terms
Here
$a=3$ (i.e. the First term)
$d=5-3=2$ ($d=$ Common difference)
(Difference between any two consecutive terms)
$n$ is number of terms
$\begin{aligned} & Sn=\dfrac{n}{2}\left[ 2a+\left( \pi -1 \right)d \right] \\\ &\Rightarrow Sn=\dfrac{n}{2}\left[ 2\times 3+\left( n-1 \right)2 \right] \\\ &\Rightarrow Sn =\dfrac{n}{2}\left[ 6+2n-2 \right] \\\ &\Rightarrow Sn=\dfrac{n}{2}\left[ 2n+4 \right] \\\ & \Rightarrow Sn =\dfrac{n}{2}\times 2\left[ 2n+2 \right] \\\ \end{aligned}$
$\therefore Sn=n\left( n+2 \right)$
So option C is correct answer for given series

Additional information:
If we know that last term then we can also use another formula for the series.
i.e. ${{S}_{n}}=\dfrac{n}{2}\left[ a+l \right]$ where $l$ is the last term
$l=a+\left( n-1 \right)d$
$a$ Is First term
$n$ Number of terms
$d$ Is common difference

Note: In these types of questions there are some patterns so that there may be Arithmetic Progression or Geometric progress. In this particular question we have Arithmetic Progress and we use the formula for the sum of the term of an Arithmetic Progression.