Q) In an AP first term is 2 and the sum of the first five terms is one- fourth of next five terms, show that 20th term is -112.

[Practice Paper 1, 2023-24, Dir of Edu, GNCT of Delhi]

Ans: 

[Here, we are given value of a, value of d is unknown. Since we have been given the relation of sum of first 5 terms and next 5 terms, we will first make equation in terms of a & d, solve them for value of d and then calculate the 20^th term]

Step1: Given that the first term is 2, ∴ a = 2
Let’s consider the common difference of the AP is d

Step 2: Let’s calculate sum of first 5 terms:
The sum of first n terms of an AP is given by: S_n = (2 a + (n – 1) d)

∴ S₅ = (2 (2) + (5 – 1) d)
= (4 + 4 d)
= 10 (1 + d)
= 10 + 10 d

Step 3: Next, let’s calculate the sum of the next 5 terms:

Sum of next 5 terms = Sum of first 10 terms – sum of first 5 terms
S_6-10 = S₁₀ – S₅
= [2 (2) + (10 – 1) d] – (10 + 10 d)
= 5 (4 + 9 d) – (10 + 10 d)
= (20 + 45 d) – (10 + 10 d)
= 20 + 45 d – 10 – 10 d
= 10 + 35 d

Step 4: Given that the sum of the first five terms is one- fourth of next five terms
∴ S₅ = S_6-10
∴ 10 + 10 d = (10 + 35 d)
∴ 40 + 40 d = (10 + 35 d)
∴ 40 + 40 d = 10 + 35 d
∴ 30 + 5 d = 0
∴ 5 d = – 30
∴ d = = – 6

Step 5: Next, let’s calculate the 20^th term:
We know that the nth term of an AP is given by: T_n = a + (n – 1) d
By substituting the values of a = 2, n = 20 and d = – 6, we get:
T₂₀ = (2) + (20 – 1) (- 6)
= 2 + 19 (- 6)
= 2 – 114
= – 112

Therefore, the value of 20^th term is – 112.

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