Question

In an arithmetic progression, the $24^{th}$ term is 100. Then, the sum of the first 47 terms of the arithmetic progression is:
A. 2300
B. 2350
C. 2400
D. 4700

Answer 1

Hint: In the above question we are provided with a term which is in AP, now we are asked to find the sum of the same series, here we need to first assume the nth term formula of the AP then equating it to the given term we would obtain the related values and hence find the summation values.
Formulae Used: The formulae for nth term of AP is:
\[ \Rightarrow {T_n} = a + (n - 1)d\]
Sum of n term of series in AP are:
\[ \Rightarrow {S_n} = \left( {\dfrac{n}{2}} \right)\left( {2a(n - 1)d} \right)\]

Complete step by step answer:
 Here the given question is of Arithmetic series, in which we need to find the sum of the 47 terms of the series, here we have the value of the $24^{th}$ term of the series, hence on solving we get:
\[ \Rightarrow {a_{24}} = a + (24 - 1)d = a + 23d\]
Now summation of n terms we have:
\[
   \Rightarrow {S_{47}} = \left( {\dfrac{{47}}{2}} \right)\left( {2a + (47 - 1)d} \right) = \left( {\dfrac{{47}}{2}} \right)\left( {2a + 46d} \right) \\
   \Rightarrow {S_{47}} = \left( {\dfrac{{47}}{2}} \right) \times 2\left( {a + 23d} \right) \\
 \]
We have:
\[ \Rightarrow {a_{24}} = a + 23d = 100\]
Putting values in the equation we have:
\[ \Rightarrow {S_{47}} = 47 \times 100 = 4700\]
Here we get the value of summation of the series.

So, the correct answer is “Option D”.

Note: Here in the given question we have to find the summation of n term of the series, here we know the formulae for summation of n term of AP and solved accordingly, any summation of series needs to be solved by its related formulae.