In arithmetic progression (A.P.), the first term is 7 and the 6th term is 22. The sum of the first 10 terms of A.P. is:

205

The correct answer is Option (2) → 205

1. Find the Common Difference ($d$)

The formula for the $n$-th term of an A.P. is:

$a_n = a + (n - 1)d$

Given:

• First term ($a$) = $7$
• 6th term ($a_6$) = $22$
• $n = 6$

Substitute the values into the formula:

$22 = 7 + (6 - 1)d$

$22 = 7 + 5d$

$15 = 5d$

$d = 3$

2. Calculate the Sum of the First 10 Terms ($S_{10}$)

The formula for the sum of the first $n$ terms is:

$S_n = \frac{n}{2} [2a + (n - 1)d]$

Substitute $n = 10, a = 7,$ and $d = 3$:

$S_{10} = \frac{10}{2} [2(7) + (10 - 1)3]$

$S_{10} = 5 [14 + 9 \times 3]$

$S_{10} = 5 [14 + 27]$

$S_{10} = 5 [41]$

$S_{10} = 205$

Conclusion

The sum of the first 10 terms of the A.P. is 205.