the sum the sum of 4th and 8th term of AP is 24 and the sum of 6th and 10th term is 44. find the first three terms of the AP.
Answers
Answer:
here,
a
4
=a+3d
a
8
=a+7d
Therefore,
a+3d+a+7d=24
2a+10d=24
a+5d=12 …… (1)
Again,
a
6
=a+5d
a
10
=a+9d
Therefore,
a+5d+a+9d=44
2a+14d=44
a+7d=22 ……. (2)
Solving equations (1) and (2), we get
d=5 and a=−13
Therefore,
a
1
=a=−13
a
2
=a+d=−13+5=−8
a
3
=a+2d=−13+10=−3
Hence, this is the required result.
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The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
Solution:
The formula for nth term of an AP is aₙ = a + (n - 1) d
Here, aₙ is the nth term, a is the first term, d is the common difference and n is the number of terms.
Given, a₄ + a₈ = 24
(a + 3d) + (a + 7d) = 24
⇒ 2a + 10d = 24
⇒ a + 5d = 12 ..... Equation(1)
Also, a₆ + a₁₀ = 44
(a + 5d ) + (a + 9d) = 44
⇒ 2a + 14d = 44
⇒ a + 7d = 22 .... Equation(2)
On subtracting equation (1) from (2), we obtain
(a + 7d ) - (a + 5d) = 22 - 12
a + 7d - a - 5d = 10
2d = 10
d = 5
By substituting the value of d = 5 in equation (1), we obtain
a + 5d = 12
a + 5 × 5 = 12
a + 25 = 12
a = - 13
The first three terms are a , (a + d) and (a + 2d)
Substituting the values of a and d , we get - 13, (- 13 + 5) and (- 13 + 2 × 5)
The first three terms of this A.P. are - 13, - 8, and - 3.