Question

if 10 times the 10th term of an AP is equal to 15 times the 15th term show that it's 25th term is 0​

Answer 1

Given

10 ( a+9d) =  15 ( a + 14d)

10a+90d = 15a + 210d

-5a  =   120d

-a  =  24d

a = -24d

Now a25   =   a + 24d

                  =  -24d  + 24d

                  = 0

Answer 2

Given:-

• 10 times the 10th term of an AP is equal to 15 times the 15th term

Prove that:-

• show that it's 25th term is 0

Proof:-

• Let the first term of the A.p as a and the common difference as d.

We know that

• an => a + (n - 1)d

For 10th term (n = 10),

• a10 => a + (10 - 1)d

=> a + 9d

For 15th term (n - 15),

• a15 => a + (15 - 1)d

=> a + 14d

Now, we are given:-

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

=> 10a + 90d = 15a + 210d

=> 90d - 210d = 15a - 10a

=> -120d = 5a

=> a = -120d/5

=> a = -24d ..........(1).

Now, the 25th term of the Ap is zero.

=> a25 = a + (25 - 1)d

=> a25 = a + 24d

=> a25 = -24d + 24d (using Eq. 1)

=> a25 = 0

Therefore, the 25th term of the given Ap is zero.

Hence, proved.