Question

The common difference of the AP whose general term ${a_n} = 2n + 1$ is

Answer 1

In this question, first of all, we will put the values of $n = 1,2,3,4,...$ in the general term ${a_n} = 2n + 1$ in order to get the Arithmetic progression after that take the difference between the terms of AP and the difference will be called as the common difference of the AP.

Complete step by step solution: We have been given general term of Arithmetic progression as ${a_n} = 2n + 1$
Now, we will put the values of $n = 1,2,3,4,...$ in ${a_n} = 2n + 1$ in order to get the AP.
For $n = 1$ we get,
${a_1} = 2\left( 1 \right) + 1$
$\Rightarrow {a_1} = 2 + 1$
$\Rightarrow {a_1} = 3$

For $n = 2$ we get,
${a_2} = 2\left( 2 \right) + 1$
$\Rightarrow {a_2} = 4 + 1$
$\Rightarrow {a_2} = 5$

For $n = 3$ we get,
${a_3} = 2\left( 3 \right) + 1$
$\Rightarrow {a_3} = 6 + 1$
$\Rightarrow {a_3} = 7$

For $n = 4$ we get,
${a_4} = 2\left( 4 \right) + 1$
$\Rightarrow {a_4} = 8 + 1$
$\Rightarrow {a_4} = 9$

Therefore, A.P. is $3,5,7,9,...$
Difference between 3 and 5 is 2.
Similarly,
Difference between 5 and 7 is 2 and
Difference between 7 and 9 is 2.

Hence, Common difference of arithmetic progression is 2.

Note: An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.
In an arithmetic progression the first term of a sequence is denoted by ${a_1}$ and the difference of successive member is d which is also called as common difference since it’s common for all the successive numbers, then the nth term of sequence $\left( {{a_n}} \right)$ is given by, ${a_n} = {a_1} + \left( {n - 1} \right)d$.