Question
4th of an AP is 14 and 8th term of an AP is 8 less than twice the 5 th term find the sum of first 25 term of AP.
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4th of an AP is 14 and 8th term of an AP is 8 less than twice the 5 th term find the sum of first 25 term of AP.
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Verified answer
EXPLANATION.
4th terms of an ap is 14.
8th terms of an ap is 8 less than twice the 5th terms.
As we know that,
General terms of an ap.
⇒ Tₙ = a + (n - 1)d.
Using this formula in this question, we get.
4th terms of an ap is 14.
⇒ T₄ = 14.
⇒ a + (4 - 1)d = 14.
⇒ a + 3d = 14. - - - - - (1).
8th terms of an ap is 8 less than twice the 5th terms.
⇒ T₈ = 8 - 2T₅.
⇒ a + (8 - 1)d = 8 - 2[a + (5 - 1)d].
⇒ a + 7d = 8 - 2[a + 4d].
⇒ a + 7d = 8 - (2a + 8d).
⇒ a + 7d = 8 - 2a - 8d.
⇒ a + 7d + 2a + 8d = 8.
⇒ 3a + 15d = 8. - - - - - (2).
We can write expression (1) as,
⇒ a + 3d = 14. - - - - - (1).
⇒ a = 14 - 3d. - - - - - (3).
Put the value of equation (3) in equation (2), we get.
⇒ 3(14 - 3d) + 15d = 8.
⇒ 42 - 9d + 15d = 8.
⇒ 6d = 8 - 42.
⇒ 6d = - 34.
⇒ 3d = - 17.
⇒ d = - 17/3.
Put the value of d = - 17/3 in equation (3), we get.
⇒ a = 14 - 3d. - - - - - (3).
⇒ a = 14 - 3 x (-17/3).
⇒ a = 14 + 17.
⇒ a = 31.
∴ First term : a = 31.
∴ Common difference : d = - 17/3.
To find : Sum of first 25 term of an ap.
As we know that,
Sum of first n terms of an ap.
⇒ Sₙ = n/2[2a + (n - 1)d].
⇒ S₂₅ = 25/2[2a + (25 - 1)d].
⇒ S₂₅ = 25/2[2a + 24d].
⇒ S₂₅ = 25/2[2(a + 12d)].
⇒ S₂₅ = 25 x [a + 12d].
Put the value in the equation, we get.
⇒ S₂₅ = 25 x [31 + 12 x (-17/3)].
⇒ S₂₅ = 25 x [31 - 4 x 17].
⇒ S₂₅ = 25 x [31 - 68].
⇒ S₂₅ = 25 x (- 37).
⇒ S₂₅ = - 925.
∴ The sum of first 25 terms of an ap is S₂₅ = - 925.
Let the first term of the AP be 'a', and the common difference be 'd'.
From the given information, we know that:
4th term = a + 3d = 14 (1)
8th term = a + 7d = 2(a + 4d) - 8 (2)
Simplifying equation (2), we get:
a - 6d = -8 (3)
Adding equations (1) and (3), we get:
2a + d = 6 (4)
Substituting equation (4) into equation (3), we get:
a = 14 - 3d
Substituting this value of 'a' into equation (4), we get:
d = 2
Substituting 'd' into equation (1), we get:
a = 8
So, the first term is 8 and the common difference is 2.
The sum of the first 25 terms of an AP can be found using the formula:
S25 = (25/2)(2a + (25-1)d)
Substituting the values of 'a' and 'd', we get:
S25 = (25/2)(2(8) + (25-1)(2))
S25 = 925
Therefore, the sum of the first 25 terms of the AP is 925.