QJEEM24Ma9241: JEE Main 2024 Mathematics - Class 11 JEEM24Ma9241 Solution: Let denote the sum of the first terms of an arithmetic…

To solve this problem, we will start by using the properties of an arithmetic progression (AP).

The sum of the first $n$ terms of an AP can be calculated using the formula: $S_n = \frac{n}{2} (2a + (n-1)d)$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, and $d$ is the common difference between the terms.

Given the information: $S_{10} = 390$

We can plug $n=10$ into the sum formula to get:

$S_{10} = \frac{10}{2} (2a + (10-1)d)$

$390 = 5(2a + 9d)$

$390 = 10a + 45d$

$78 = 2a + 9d \quad .........\text{(1)}$

Next, we're given the ratio of the tenth term ( $T_{10}$ ) to the fifth term ( $T_5$ ): $\frac{T_{10}}{T_5} = \frac{15}{7}$

The $n$ th term of an AP is given by:

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

So, for the tenth term: $T_{10} = a + (10-1)d = a + 9d$

And for the fifth term:

$T_5 = a + (5-1)d = a + 4d$

Now we can write the ratio as:

$\frac{a + 9d}{a + 4d} = \frac{15}{7}$

$7(a + 9d) = 15(a + 4d)$

$7a + 63d = 15a + 60d$

$63d - 60d = 15a - 7a$

$3d = 8a \quad .........\text{(2)}$

Now we have two equations (1) and (2):

$78 = 2a + 9d \quad \text{(1)}$

$3d = 8a \quad \text{(2)}$

We can solve these equations simultaneously.

From equation (2):

$d = \frac{8}{3}a$

Plugging this back into (1):

$78 = 2a + 9\left(\frac{8}{3}a\right)$

$78 = 2a + 24a$

$78 = 26a$

$a = 3$

Now we can find $d$ :

$d = \frac{8}{3}a$

$d = \frac{8}{3} \times 3$

$d = 8$

Now we can find $S_{15}$ and $S_5$ using the formula for the sum of an AP.

For $S_{15}$ :

$S_{15} = \frac{15}{2} (2 \cdot 3 + (15-1) \cdot 8)$

$S_{15} = \frac{15}{2} (6 + 14 \cdot 8)$

$S_{15} = \frac{15}{2} (6 + 112)$

$S_{15} = \frac{15}{2} \cdot 118$

$S_{15} = 15 \cdot 59$

$S_{15} = 885$

For $S_5$ :

$S_5 = \frac{5}{2} (2 \cdot 3 + (5-1) \cdot 8)$

$S_5 = \frac{5}{2} (6 + 4 \cdot 8)$

$S_5 = \frac{5}{2} (6 + 32)$

$S_5 = \frac{5}{2} \cdot 38$

$S_5 = 5 \cdot 19$

$S_5 = 95$

The difference $S_{15} - S_{5}$ is:

$S_{15} - S_{5} = 885 - 95$

$S_{15} - S_{5} = 790$

Therefore, the correct answer is Option C, which is 790.