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 nth term \( T_n \) of an A.P. can be calculated using the formula:
\[ T_n = a + (n - 1) \cdot d \]
where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

### Step 2: Set up the equation for the 22nd term
Given that the 22nd term \( T_{22} = 149 \), we can write:
\[ T_{22} = a + (22 - 1) \cdot d = 149 \]
Substituting \( d = 7 \):
\[ a + 21 \cdot 7 = 149 \]

### Step 3: Simplify the equation
Calculate \( 21 \cdot 7 \):
\[ 21 \cdot 7 = 147 \]
Now substitute this back into the equation:
\[ a + 147 = 149 \]

### Step 4: Solve for the first term \( a \)
To find \( a \), subtract 147 from both sides:
\[ a = 149 - 147 = 2 \]

### Step 5: Use the formula for the sum of the first n terms of an A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
For our case, we need to find \( S_{22} \):
\[ S_{22} = \frac{22}{2} \cdot (2a + (22 - 1) \cdot d) \]

### Step 6: Substitute the known values into the sum formula
Substituting \( a = 2 \), \( n = 22 \), and \( d = 7 \):
\[ S_{22} = 11 \cdot (2 \cdot 2 + 21 \cdot 7) \]

### Step 7: Calculate \( 2a + (n - 1) \cdot d \)
Calculate \( 2a \):
\[ 2 \cdot 2 = 4 \]
Calculate \( 21 \cdot 7 \):
\[ 21 \cdot 7 = 147 \]
Now substitute these values back into the equation:
\[ S_{22} = 11 \cdot (4 + 147) = 11 \cdot 151 \]

### Step 8: Final calculation of the sum
Now, calculate \( 11 \cdot 151 \):
\[ S_{22} = 1661 \]

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

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