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Five numbers are in A.P. with common difference ≠ 0. If 1st, 3rd and 4th terms are in G.P., then which term is always zero?

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Let the five numbers in A.P. be a – 2d, a – d, a, a + d, a + 2d. 

Since 1st, 3rd and 4th terms are in G.P. 

(t3)2 = t1 . t4 

⇒ a2 = (a – 2d) (a + d) 

⇒ a2 = a2 – ad – 2d2 

⇒ – ad = 2d2 ⇒ a = – 2d            ( d ≠ 0) 

⇒ a + 2d = 0 ⇒ t5 = 0

∴ 5th term is always zero.

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