The Total solution for NCERT class 6-12
If 10 times the 10th term of an A.P. is equal to 15 timesthe 15th term, show that the 25th term of theA.P. is Zero.
Given:
10 times the 10th termof an A.P. is equal to 15 times the 15th term
So, 10a10 =15a15
We need to prove: a25 =0
We know, an =a + (n – 1) d [where a is first term or a1 and d is commondifference and n is any natural number]
When n = 10:
a10 =a + (10 – 1)d
= a + 9d
When n = 15:
a15 =a + (15 – 1)d
= a + 14d
When n = 25:
a25 =a + (25 – 1)d
= a + 24d ………(i)
According to question:
10a10 =15a15
10(a + 9d) = 15(a +14d)
10a + 90d = 15a + 210d
10a – 15a + 90d – 210d= 0
-5a – 120d = 0
-5(a + 24d) = 0
a + 24d = 0
a25 =0 [From (i)]
Hence Proved.