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**Hint:**Use the formula of Arithmetic progression sequence for the nth terms that is \[{a_n} = a + \left( {n - 1} \right)d\] where, a initial term of the AP and d is the common difference of successive numbers. Calculate the value of a. We use the formula of the sum of n terms in Arithmetic progression that is \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\]. Calculate the sum of the AP, \[{S_n}\].

**Complete step by step solution:**

**Given data:**The ${22^{nd}}$ term that is given for an arithmetic progression is 149.

Common difference is \[d = 7\]

Now, we know about the Arithmetic progression sequence for the nth terms is given by the following expression:

\[{a_n} = a + \left( {n - 1} \right)d\]

Here, the first term of the arithmetic progression sequence is $a$.

Now, calculate the value of $a$. Substitute the value of d = 7,n = 22 and ${a_n} = 149$ in \[{a_n} = a + \left( {n - 1} \right)d\].

149 = a + (22- 1)7

149 = a + 147

a = 149 - 147

= 2

Now, we know about the formula of the sum of n terms in Arithmetic progression is given by the following expression:

\[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\]

Simplify the above equation by substituting \[{a_n} = a + \left( {n - 1} \right)d\].

\[{S_n} = \dfrac{n}{2}\left[ {a + {a_n}} \right]\]

Now, calculate the value of ${S_n}$ by substituting $n = 23$, $a = 2$ and $a_n = 149$ in the expression for the sum of the Arithmetic progression \[{S_n} = \dfrac{n}{2}\left[ {a + {a_n}} \right]\].

${S_{22}} = \dfrac{{22}}{2}\left[ {2 + 149} \right]\\

= 11\left[ {151} \right]\\

= 1,661

$

**Hence, the sum of the first 22 terms of an Arithmetic progression is \[{S_{22}} = 1,661\].**

**Note:**The general equation of the Arithmetic progression is \[a,a + d,a + 2d,a + 3d,...\], where a is initial term of the AP and d is the common difference of successive numbers. Make sure use the formula of the sum of n terms in Arithmetic progression that is \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\] and use the Arithmetic progression sequence for the nth terms that is \[{a_n} = a + \left( {n - 1} \right)d\].

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