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State true or false.

The 19th term of an AP is equal to 3 times its 6th term. If its 9th term is 19 then the AP series is 3,5,7,9.....

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Detailed Solution

Given that the 19th term of an AP is equal to 3 times its 6th term and the 9th terms of A.P. is 19.
Formula used for

n t h

term of A.P.:

a n = a + n − 1 d

∵ a 19 = 3 a 6

Applying the formula of nth term of A.P.,

a 19 = a + (19 − 1) d a 6 = a + (6 − 1) d ∴ a + (19 − 1) d = 3 {a + (6 − 1) d} ⇒ a + 18 d = 3 a + 15 d ⇒ a = 3 d 2 − − − − − (1)

∵ a 9 = 19

∴ a + (9 − 1) d = 19 ⇒ a + 8 d = 19 − − − − − (2)

Put the value of equation (1) to equation (2):

∴ 3 d 2 + 8 d = 19 ⇒ 3 d + 16 d = 38 ⇒ 19 d = 38 ⇒ d = 2

Put the value of d in equation (1)

∴ a = 3 × 2 2 ⇒ a = 3

Put the value of a and d in the general series,

∴ (3), [3 + 2], [3 + 2 × 2], [3 + 3 × 2], … … or (3), (5), (3 + 4), (3 + 6), … … or (3), (5), (7), (9), …

Therefore, the A.P. series is 3,5,7,9....
Thus, the given statement is true.
Hence, option (1) is correct.

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