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Hint- Here, we will be using the formula for the sum of first n terms of an arithmetic progression (AP) which is ${{\text{S}}_n} = \dfrac{n}{2}\left[ {2{a_1} + \left( {n - 1} \right)d} \right]$ and in order to get the sum of the first 22 terms of the given AP, we will evaluate ${{\text{S}}_{22}}$ by substituting n=22.

Complete step-by-step answer:

The given AP series is 1,4,7,10,….

Clearly, the first term of the given series is ${a_1} = 1$

Here, common difference is $

d = 4 - 1 \\

\Rightarrow d = 3 \\

$

As we know that the sum of first n terms of an AP series having first term as ${a_1}$ and common difference as d is given by

${{\text{S}}_n} = \dfrac{n}{2}\left[ {2{a_1} + \left( {n - 1} \right)d} \right]$

Now for the sum of the first 22 terms of the given AP series, we will put n=22, ${a_1} = 1$ and d=3 in the above formula.

$

{{\text{S}}_{22}} = \dfrac{{22}}{2}\left[ {2\left( 1 \right) + 3\left( {22 - 1} \right)} \right] \\

\Rightarrow {{\text{S}}_{22}} = 11\left[ {2 + 63} \right] \\

\Rightarrow {{\text{S}}_{22}} = 715 \\

$

Therefore, the sum of the first 22 terms of the given AP series is 715.

Note- An arithmetic progression (AP) is the series which will have the same common difference between any two consecutive terms like in the given series the common difference is constant between any two consecutive terms i.e., 3. For any general AP, the formula for ${n^{th}}$ term of this AP is given by ${a_n} = {a_1} + \left( {n - 1} \right)d$.

Complete step-by-step answer:

The given AP series is 1,4,7,10,….

Clearly, the first term of the given series is ${a_1} = 1$

Here, common difference is $

d = 4 - 1 \\

\Rightarrow d = 3 \\

$

As we know that the sum of first n terms of an AP series having first term as ${a_1}$ and common difference as d is given by

${{\text{S}}_n} = \dfrac{n}{2}\left[ {2{a_1} + \left( {n - 1} \right)d} \right]$

Now for the sum of the first 22 terms of the given AP series, we will put n=22, ${a_1} = 1$ and d=3 in the above formula.

$

{{\text{S}}_{22}} = \dfrac{{22}}{2}\left[ {2\left( 1 \right) + 3\left( {22 - 1} \right)} \right] \\

\Rightarrow {{\text{S}}_{22}} = 11\left[ {2 + 63} \right] \\

\Rightarrow {{\text{S}}_{22}} = 715 \\

$

Therefore, the sum of the first 22 terms of the given AP series is 715.

Note- An arithmetic progression (AP) is the series which will have the same common difference between any two consecutive terms like in the given series the common difference is constant between any two consecutive terms i.e., 3. For any general AP, the formula for ${n^{th}}$ term of this AP is given by ${a_n} = {a_1} + \left( {n - 1} \right)d$.

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