If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.
Given: 10 × a10 = 15 × a15
To show : a25 = 0
Consider 10 × a10 = 15 × a15
⇒ 10 [a + (10 - 1)d] = 15 [a + (15 - 1)d]
⇒ 10a + 90d = 15a + 210d
⇒ - 5 a = 120 d
⇒ a = - 24d ………………(1)
Now, a25 = a + (25 - 1)d
⇒ a25 = a + 24d
⇒ a25 = - 24d + 24d (from equation 1)
⇒ a25 = 0
Hence, proved.