# The sum of 5^th and 9^th terms of an AP is 30. If its 25^th term is three times its 8^th term, find the AP.

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The sum of 5th and 9th terms of an AP is 30. If its 25th term is three times its 8th term, find the AP.

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a5 +a= 30

(a +4d) + (a +8d) = 30

2a +12d = 30

a + 6d = 15 (i)

Acc. To question,

a25 = 3a8

a + 24d = 3(a +7d)

3d = 2a

3d = 2(15 - 6d) [from(i)]

15d = 30

d = 2

Putting the value of d in (i),

a = 15 – 6(2) = 3

a1 = a = 3

a2 = a +d = 3 +2 = 5

a3 = a +2d = 3 + 2(2) = 7

Hence, the A.P. is 3, 5, 7, 9,.......

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HELLO,

Let the AP is $a_1, a_2, a_3,...a_n$ with common difference d.

Given in the question,

$a_5 + a_9= 30$

$a_1+4d+a_1+8d=30$

$2a_1+12d=30$

$a_1+6d=15$                         ------------- 1

Now again given in the question,

$a_{25}=3a_8$

$a_1+24d= 3(a_1+7d)$

$a_1+24d= 3a_1+21d$

$2a_1-3d=0$         -------------- 2

Now solving equation 1 and 2 we get     ( $2* eqn 1 - eqn2$),

$15d=30$

d=2 and $a_1=3$

hence, $a_1=3, a_2= 3+2, a_3= 3+2+2 ...$

So, the required AP is 3,5,7,9 ...

I HOPE YOU WILL UNDERSTAND.