The 7th and 9th terms of an arithmetic progression are 10 and 11, res…

The correct answer is Option (2) → 14

To find the 15th term of the Arithmetic Progression (A.P.), we use the general formula for the $n^{th}$ term:

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

where $a$ is the first term and $d$ is the common difference.

1. Set up the equations

We are given the 7th and 9th terms:

1. 7th term ($a_7$): $a + 6d = 10$
2. 9th term ($a_9$): $a + 8d = 11$

2. Solve for the common difference ($d$)

Subtract the first equation from the second:

$(a + 8d) - (a + 6d) = 11 - 10$

$2d = 1$

$d = \frac{1}{2} = 0.5$

3. Solve for the first term ($a$)

Substitute $d = 0.5$ into the first equation:

$a + 6(0.5) = 10$

$a + 3 = 10$

$a = 7$

4. Find the 15th term ($a_{15}$)

Now, substitute $a = 7$, $d = 0.5$, and $n = 15$ into the general formula:

$a_{15} = 7 + (15 - 1)(0.5)$

$a_{15} = 7 + 14(0.5)$

$a_{15} = 7 + 7$

$a_{15} = 14$