Question

The first term of an arithmetic progression is 7 and the ratio of 8 and 15 term is 4 is 27 find the sum of first 25 terms in the progression

Answer 1

Step-by-step explanation:

Given:The first term of an arithmetic progression is 7 and the ratio of 8 and 15 term is 4 is 27 find the sum of first 25 terms in the progression

To find:Find the sum of first 25 terms in the progression

Solution:

Let the first term of A.P. is a= 7

nth term of AP :

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

Thus,

$a_8 = a + 7d \\ \\ a_{15} = a + 14d$

ATQ

$\frac{a_8}{a_{15}} = \frac{a + 7d}{a + 14d} = \frac{4}{27}$

put the value of a

$\frac{7 + 7d}{7 + 14d} = \frac{4}{27} \\ \\ \frac{1 + d}{1 + 2d} = \frac{4}{27} \\ \\ 27 + 27d = 4 + 8d \\ \\ 27d - 8d = 4 - 27 \\ \\ 19d = - 23 \\ \\ d = \frac{ - 23}{19}$

To find the sum of first 25 terms,put the value of a,d and n in formula

$\bold{S_n = \frac{n}{2} [2a + (n - 1)d] }\\ \\ S_{25} = \frac{25}{2} (2 \times 7 + 24 \times \frac{ - 23}{19} ) \\ \\ S_{25} = \frac{25}{2} (14 - 29.05 ) \\ \\ S_{25} = 12.5 \times ( - 15.05) \\ \\\bold{ S_{25} = -188.16 }$

Hope it helps you.

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