Question

If the sum of first n terms of an A.P. is given by $3{n^2} - n$, then find its ${25^{th}}$ term.

Answer 1

Hint: We will first establish a relation between the sum of terms of an A.P. to the ${n^{th}}$ term of the A.P. Now after establishing the formula, we will put in the required value of n to get our answer.

Complete step-by-step answer:
Let us first get to know what an A.P. is:-
An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Now, we should notice that if we subtract the sum of n terms from the sum of n + 1 terms, we will get the ${n^{th}}$ term.
We can see this as:-
Consider ${s_{n + 1}} = {a_1} + {a_2} + ...... + {a_{n + 1}}$ and ${s_n} = {a_1} + {a_2} + ...... + {a_n}$.
Now on subtracting them, we will have:-
\[{s_{n + 1}} - {s_n} = {a_1} + {a_2} + ...... + {a_{n + 1}} - ({a_1} + {a_2} + ...... + {a_n})\]
This is equivalent to:-
\[{s_{n + 1}} - {s_n} = {a_1} + {a_2} + ...... + {a_{n + 1}} - {a_1} - {a_2} - ...... - {a_n}\]
Simplifying this by clubbing the same terms, we will get:-
\[{s_{n + 1}} - {s_n} = {a_{n + 1}}\] (Because rest all of the terms get cancelled out by each other)
Now, we have got the required formula we will use in this solution:-
\[{s_{n + 1}} - {s_n} = {a_{n + 1}}\].
We need to find ${25^{th}}$ term. So, let us put n = 24.
So, we will get:-
\[ \Rightarrow {a_{24 + 1}} = {s_{24 + 1}} - {s_{24}}\]
Putting the value of ${s_n}$ as $3{n^2} - n$. We will get:-
\[ \Rightarrow {a_{25}} = {s_{25}} - {s_{24}} = 3{(25)^2} - 25 - \{ 3{(24)^2} - 24\} \]
\[ \Rightarrow {a_{25}} = 3 \times 625 - 25 - \{ 3 \times 576 - 24\} \]
\[ \Rightarrow {a_{25}} = 1850 - 1704 = 146\].
Hence, the required term is 146.
Hence, the answer is 146.

Note: The students must note that if we take the sum of only the first term, it will be the term itself only. Therefore, we have ${a_1} = {s_1}$.
The students might make the mistake of putting in the value of n directly to get the answer but you must remember that this method will give you the sum of all the terms up to that number, not that term itself.
Fun fact:- According to an anecdote, young Carl Friedrich Gauss reinvented the method of summation of terms to compute the sum 1 + 2 + 3 + ... + 99 + 100 for a punishment in primary school.