Question

The set ${S_1},{S_2},{S_3}$… are given by:
\Rightarrow {S_3} = \left\\{ {\dfrac{4}{3},\dfrac{7}{3},\dfrac{{10}}{3}} \right\\},{S_2} = \left\\{ {\dfrac{3}{2},\dfrac{5}{2}} \right\\},{S_1} = \left\\{ {\dfrac{2}{1}} \right\\},{S_4} = \left\\{ {\dfrac{5}{4},\dfrac{9}{4},\dfrac{{13}}{4},\dfrac{{17}}{4}} \right\\}
Then the sum of the numbers in set ${S_{25}}$, is:
A. 320
B. 322
C. 324
D. 326

Answer 1

Here in the given question we are provided with a series, in every set, here we first we need to find the relation between the term of the series and then we can do summation for the given number of terms, in order to find the required sum as asked in the question.

Formulae Used: Sum of n term of series which is in AP:
$\Rightarrow {S_n} = \dfrac{n}{2}\left[ {2 \times a + (n - 1)} \right]$
Common difference of the series in AP= Second term-first term

Complete step by step answer:
Here in the given question we need to first see for the relation between the terms, then check for the required series, in order to find the sum, as asked in the question, on solving we get:
The required series, can be written as, as per the pattern in the given series we have:
\Rightarrow {S_{25}} = \left\\{ {\dfrac{{26}}{{25}},\dfrac{{51}}{{25}},\dfrac{{76}}{{25}},...upto\,25\,terms} \right\\}
Here we have observed that the respective series are in arithmetic series, as here the common difference between the series are same, as we see:

$$\Rightarrow Common\,difference\,{S_2} = 1 \\\ \Rightarrow Common\,difference\,{S_3} = 1 \\\ \Rightarrow Common\,difference\,{S_4} = 1 \\\$$

So here the common difference for the required series will be:
$\Rightarrow Common\,difference\,{S_{25}} = 1$

And:
$\Rightarrow First\,term = \dfrac{{26}}{{25}},\,n = 25$
Finding sum with the help of first term, common difference and the number of terms for an arithmetic series we have:

$$\Rightarrow {S_n} = \dfrac{n}{2}\left[ {2 \times a + (n - 1)d} \right] \\\ \Rightarrow {S_{25}} = \dfrac{{25}}{2}\left[ {2 \times \dfrac{{26}}{{25}} + (25 - 1)} \right] \\\ \Rightarrow {S_{25}} = \dfrac{{25}}{2}\left[ {\dfrac{{52}}{{25}} + 24} \right] = 326 \\\$$

Here we got the required sum of the given series.

So, the correct answer is “Option D”.

Note: Here in such a type of question where a specific series is provided to solve with, in such a condition first the series is needed to be recognized in order to solve for the solution, here we first see that the required series is in AP, and then go for the solution.