Question

if 10th times the 10th term of an AP is equal to 15 times the 15th term find the 25th term​

Answer 1

Answer:

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Step-by-step explanation:

We know that the n

th

term of the arithmetic progression is given by a+(n−1)d

Given that the 10 times the 10

th

term is equal to 15 times the 15

th

term

Therefore, 10(10

th

term)=15(15

th

term)

⟹10(a+(10−1)d)=15(a+(15−1)d)

⟹10(a+9d)=15(a+14d)

⟹10a+90d=15a+210d

⟹15a−10a=90d−210d

⟹5a=−120d

⟹a=−24d ------(1)

The 25

th

term is a+(25−1)d=a+24d=−24d+24d=0

Therefore, the 25

th

term of the A.P. is zero

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Answer 2

$Let \: \: a = \: first \: term$

$and \: \: d = difference$

$10 \times a_{10} = 15 \times a_{15} \: (given)$

$10 = (a + 9d) = 15(a + 14d)$

$a_{n} = a + (n - 1)d$

$2(a + 9d) = 3(a + 14 \: d)$

$2a + 18d = 3a + 42d$

$2a - 3a = 42d - 18d$

$- a = - 24d$

$a = - 24d$

$a + 24d = 0$

$a + (25 - 1)d = 0$

$a_{25} = 0$

$25th \: Term \: of \: an \: A.P \: is \: Zero .$