Find the sum of first 22 terms of an A.P. in which d = 7 and 22 nd term is 149.

Answer

To find the sum of the first 22 terms of an arithmetic progression (A.P.) where the common difference \( d = 7 \) and the 22nd term is \( 149 \), we can follow these steps:

### Step 1: Identify the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1)d \]
where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( n \) is the term number,
- \( d \) is the common difference.

### Step 2: Substitute the known values into the formula.
In this case, we know:
- \( n = 22 \),
- \( a_{22} = 149 \),
- \( d = 7 \).

Substituting these values into the formula gives:
\[ 149 = a + (22 - 1) \cdot 7 \]

### Step 3: Simplify the equation.
Calculating \( 22 - 1 \) gives \( 21 \):
\[ 149 = a + 21 \cdot 7 \]
Calculating \( 21 \cdot 7 \) gives \( 147 \):
\[ 149 = a + 147 \]

### Step 4: Solve for \( a \).
To find \( a \), we rearrange the equation:
\[ a = 149 - 147 \]
Calculating this gives:
\[ a = 2 \]

### Step 5: Use the formula for the sum of the first \( n \) terms of an A.P.
The formula for the sum of the first \( n \) terms \( S_n \) is:
\[ S_n = \frac{n}{2} (a + l) \]
where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( l \) is the last term (which in this case is \( a_{22} = 149 \)).

### Step 6: Substitute the known values into the sum formula.
Here, we have:
- \( n = 22 \),
- \( a = 2 \),
- \( l = 149 \).

Substituting these values gives:
\[ S_{22} = \frac{22}{2} (2 + 149) \]

### Step 7: Simplify the sum expression.
Calculating \( \frac{22}{2} \) gives \( 11 \):
\[ S_{22} = 11 \cdot (2 + 149) \]
Calculating \( 2 + 149 \) gives \( 151 \):
\[ S_{22} = 11 \cdot 151 \]

### Step 8: Calculate the final sum.
Calculating \( 11 \cdot 151 \) gives:
\[ S_{22} = 1661 \]

### Final Answer:
The sum of the first 22 terms of the A.P. is \( 1661 \).

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