Question

Find the sum $2,3\dfrac{1}{4},4\dfrac{1}{2},$…..To $20$terms.

Answer 1

Hint:-Here, this is a term of $AP$so we apply summation of $n$terms of$AP$.

Given series is$2,3\dfrac{1}{4},4\dfrac{1}{2},$…..Up to $20$terms.
The given series can be written as$\dfrac{8}{4},\dfrac{{13}}{4},\dfrac{{18}}{4}$,……to $20$ terms.
Take $\dfrac{1}{4}$ common we get
$\Rightarrow \dfrac{1}{4}(8 + 13 + 18 + ......)$ To $20$ terms.
Observe that ${\text{8,13,18}}$…… is in Arithmetic Progression with first term $8$and$5$as common difference
So, the sum of first$20$ terms in that series will be
$= \dfrac{1}{4}\left( {\dfrac{{20}}{2}\left( {2 \times 8 + (20 - 1) \times 5} \right)} \right)$ $\because {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
Here $n =$ number of terms, $a =$ first term and $d =$ common difference.
${\text{ = }}\dfrac{1}{4}(1110) = \dfrac{{555}}{2}$ Answer.

Note: - Whenever such type of series is given first convert into simple form and then find which series is this. Then apply the formula of that series to get the answer.