Question

The sum of the 5th and the 9th term of an ap is 30 if its 25th term is three times its 8th term find the ap

Answer 1

Solution:

Given sum of 5th term and 9th term of AP=30

a+4d + a+8d =30

2a + 12 d = 30 ---equation (1)

Given that 25 term is three times of 3rd term in AP

25th term= 3 (8th term)

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

a+ 24d = 3a+ 21d
24d - 21 d= 3a- a
3d = 2a --- equation 2

Substitute the value of 2d in equation (1) we get

2a+ 12d = 30

3d +12d =30

15d =30

d = 2

2a = 3d
2a= 3x 2
2a= 6
a=3

THEREFORE AP is 3, 5,7, 9..

Answer 2

Solution :-

The sum of 5th and 9th term is 30.

⇒ a + 4d + a + 8d = 30

⇒ 2a + 12d = 30 ..................(1)

And,

Its 25th term is 3 times its 8th term.

⇒ a + 24d = 3(a + 7d)

⇒ a + 24d = 3a + 21d

⇒ a + 24d - 3a - 21d = 0

⇒ - 2a + 3d = 0 ...................(2)

Adding equation (1) and (2).

⇒ 2a +12d = 30 
  - 2a + 3d =   0
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           15d = 30
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⇒ 15d = 30

⇒ d = 30/15

⇒ d = 2

So, common difference, 'd' is 2.

Putting the value of d = 2 in the equation (1).

2a + 12d = 30

⇒ 2a + (12*2) = 30

⇒ 2a = 24 = 30

⇒ 2a = 30 - 24

⇒ 2a = 6

⇒ a = 6/2

⇒ a = 3

So, the first term of the required AP is 3

So, the required AP is 3, 5, 7, 9, 11, 13, 15, 19,.............

Answer.