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.

Given:

10 times the 10th term of an A.P. is equal to 15 times the 15th term.

To do:

We have to show that the 25th term of the A.P. is zero.

Solution:

Let the required A.P. be $a, a+d, a+2d, ......$

Here,

$a_1=a, a_2=a+d$ and Common difference $=a_2-a_1=a+d-a=d$

We know that,

$a_n=a+(n-1)d$

Therefore,

$a_{10}=a+(10-1)d$

$=a+9d$

$10\times a_{10}=10(a+9d)$.....(i)

$a_{15}=a+(15-1)d$

$=a+14d$

$15\times a_{15}=15(a+14d)$....(ii)

From (i) and (ii), we get,

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

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

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

$3a-2a+42d-18d=0$

$a+24d=0$

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

$\Rightarrow a_{25}=a+(25-1)d=0$

Hence proved.

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Updated on: 10-Oct-2022

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