Question

If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then what is the sum of the first ten terms?
1. -500
2. -250
3. 500
4. 250

Answer 1

Correct Answer - Option 2 : -250

Concept:

Let us consider sequence a₁, a₂, a₃ …. a_n is an A.P.

Common difference “d”= a₂ – a₁ = a₃ – a₂ = …. = a_n – a_{n – 1}

n^th term of the A.P. is given by a_n = a + (n – 1) d

Sum of the first n terms (S) =  \(\frac{n}{2}[2a+(n-1)d)]\)

Also, S =  (n/2)(a + l)

Where,
a = First term,
d = Common difference,
n = number of terms,
a_n = n^th term,
l = Last term

Calculation:

Given, a₀ = 2      ...(1)

S₅ = \(1\over4\)(S₁₀ - S₅)

⇒ 4S5 = S10 - S5

⇒ 4S₅ + S₅ = S₁₀

⇒ 5S₅ = S₁₀

⇒ 5 × \(5\over2\)[2a0 + (n₅ - 1)d] = \(10 \over2\) [2a0 + (n₁₀ - 1)d]

[∵ n₅ = 5 & n₁₀ = 10]

⇒ 5 × \(5\over2\)[a₀ + a₀ + 4d] = \(10 \over2\) [a0 + a0 + 9d]

⇒ 5 × [2a0 + 4d] = 2 × [2a0 + 9d]

⇒ 10a₀ + 20d = 4a₀ + 18d

⇒ 6a0 = 2d

⇒ a₀ = \(\rm-d\over3\)

∴ d = -3a₀ = - 3 × 2 = - 6

Hence,

S₁₀ = \(10 \over2\) [a0 + a0 + 9d]

⇒ 5[2a₀ + 9d]

⇒ 5[4 - 54]

As a₀ = 2, d = - 6

⇒ S10 = 5 × (-50) = - 250