Question

If the sum of the first 8 terms of AP is 136 and that of first 15 terms is 465 then find the sum of first 25 terms

Answer 1

Answer:

$S_{25} =1275$

Step-by-step explanation:

To find the sum of first 25 terms;

If the sum of the first 8 terms of AP is 136 and that of first 15 terms is 465.

We know that sum of first n terms of an AP having first term 'a' and common difference 'd'

$S_n = \frac{n}{2} \Big(2a + (n - 1)d\Big) \\ \\ S_8 = \frac{8}{2} (2a + (8- 1)d) \\ \\ S_8 = 4(2a + 7d) \\ \\ 136 = 4(2a + 7d) \\ \\ \frac{136}{4} = 2a + 7d \\ \\ 2a + 7d = 34.....eq1$

By the same way

$S_{15} = \frac{15}{2} (2a + (15- 1)d) \\ \\ S_{15} = \frac{15}{2} (2a + 14d) \\ \\ 465 \times 2 = 15(2a + 14d) \\ \\ \frac{465 \times 2}{15} = 2a + 14d \\ \\ 2a + 14d = 62.....eq2$

Now subtract both equations and find the value of a and d

$2a + 7d = 34 \\ 2a + 14d = 62 \\ - - - - - - - \\ - 7d = - 28 \\ \\ d = 4$

Thus

$2a + 7d = 34 \\ \\ 2a + 28 = 34 \\ \\ 2a = 34 - 28 \\ \\ a = 3$

So,sum of first 25 terms

$S_n = \frac{n}{2} \Big(2a + (n - 1)d\Big) \\ \\ S_{25} = \frac{25}{2} (6 + 24 \times 4) \\ \\ S_{25}= \frac{25}{2}(102) \\ \\S_{25} =1275$

Hope it helps you.

Answer 2

Answer:

Step-by-step explanation: