If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

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Solution

Given:
$10 a_{10} = 15 a_{15}$
$\Rightarrow 10 [a + (10 - 1) d] = 15 [a + (15 - 1) d] \Rightarrow (10 a + 90 d) = 15 (a + 14 d) \Rightarrow 10 a + 90 d = 15 a + 210 d \Rightarrow 0 = 5 a + 120 d \Rightarrow 0 = a + 24 d \Rightarrow a = - 24 d \dots \dots (i)$
To show:
$a_{25} = 0$
$\Rightarrow L H S : a_{25} = a + (25 - 1) d = a + 24 d = - 24 d + 24 d [From (i)] = 0 = R H S$
Hence proved.

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