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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

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Correct Answer - Option 2 : -250

Concept:

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

Common difference “d”= a2 – a1 = a3 – a2 = …. = an – an – 1

nth term of the A.P. is given by an = 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,
an = nth 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

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