The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms
Solution:
Given, the sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167.
The sum of the first ten terms of the AP is 235.
We have to find the sum of the first 20 terms.
The sum of the first n terms of an AP is given by
Sₙ = n/2[2a + (n-1)d]
When n = 10,
S₁₀ = 10/2[2a + (10 - 1)d]
235 = 5[2a + 9d]
2a + 9d = 47 ---------------------- (1)
When n = 5,
S₅ = 5/2[2a + (5 - 1)d]
= 5/2[2a + 4d]
Taking out common term,
= (5/2)2[a + 2d]
= 5(a + 2d)
S₅ = 5a + 10d
When n = 7,
S₇ = 7/2[2a + (7 - 1)d]
= 7/2[2a + 6d]
Taking out common term,
= (7/2)2[a + 3d]
= 7(a + 3d)
S₇ = 7a + 21d
Given, S₅ + S₇ = 167
5a + 10d + 7a + 21d = 167
12a + 31d = 167 -------------- (2)
Multiplying (1) by 6,
12a + 54d = 282 -------------- (3)
Subtracting (3) and (2),
12a + 54d - (12a+ 31d) = 282 - 167
12a + 54d - 12a - 31d = 115
23d = 115
d = 115/23
d = 5
Put d = 5 in (1),
2a + 9(5) = 47
2a + 45 = 47
2a = 47 - 45
2a = 2
a = 1
To find the sum of the first 20 terms,
S₂₀ = 20/2[2(1) + (20 - 1)5]
= 10[2 + 19(5)]
= 10(2 + 95)
= 10(97)
= 970
Therefore, the sum of the first twenty terms is 970.
✦ Try This: The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 150. If the sum of the first ten terms of this AP is 355, find the sum of its first twenty terms
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 5
NCERT Exemplar Class 10 Maths Exercise 5.4 Problem 1
The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms
Summary:
The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, the sum of its first twenty terms is 970.
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