Question

Sum of n terms of an A.P. is \[{n^2} + 2n\] . Find the first term and the common difference.

Answer 1

Hint: Here, the nth term of an A.P. is given, so by putting different values of n we can find different terms. In this question, find first term and second term by putting n = 1 and n = 2 respectively. Then to find common differences subtract the first term from second term.

Complete step-by-step answer:
A.P.: Arithmetic progression is a sequence of numbers having the same common difference between the two consecutive terms. For example if we write numbers as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 this is an A.P. with the first term as 1 and common difference as 1. We can also write general terms as n, now by putting different values of n we will get the value of a particular term. In this problem the general term given is \[{n^2} + 2n\] .
We get first term by putting n = 1.
First term = \[{1^2} + 2 \times 1 = 1 + 2 = 3\]
We get second term by putting n = 2.
Second term = \[{2^2} + 2 \times 2 = 4 + 4 = 8\]
Common difference = Second term – First term
Common difference = 8 – 3 = 5
Thus, first term = 3 and common difference = 5.
So, the correct answer is “ first term = 3 and common difference = 5”.

Note: In these types of questions, where the formula of nth term is given, we can write A.P. by putting values as n = 1, n = 2, n = 3, … etc. For common difference means second term – first term or third term – second term or in general common difference of an A.P. is nth term – (n – 1)th term.