Q. 34.0( 155 Votes )

# In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20^{th} term is –112.

Answer :

Given, first term a = 2

Let d be the common difference

Let S_{n} denote sum of n terms,

t_{n} denote n^{th} term.

t_{n} = a + (n – 1)d

S_{1} = Sum of first five terms.

S_{2} = Sum of next five terms.

S_{1} = 10 × (1 + d)

Sum of next 10 terms = Sum of first 10 terms - Sum of first 5 terms

S_{2} = 20 + 45d – 10 – 10d = 10 + 35d

Also, S_{1} = 1/4 S_{2}

⇒ 10 + 10d = 1/4 (10 + 35d)

⇒ 40 + 40d = 10 + 35d

⇒ 30 = -5d

⇒ d = -6

t_{20} = 2 + (20 – 1) × (-6)

t_{20} = 2 – 19 × 6 = 2 – 114

t_{20} = -112

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