Question

If 5 times the 5th term of an AP is equal to 8 times its 8th term, show that the 13th term is zero.

Answer 1

let of the AP
first term= a
common difference=d
ATQ
5[a+(5-1)d]=8[a+(8-1)d]
5(a+4d)=8(a+7d)
5a + 20d = 8a + 56d
3a = 36d
a= -12d
Now 13th term
a+(13-1)d
=-12d+12d
=0
(proved)

Answer 2

Given,
5a₅ = 8a₈
Let a and d be the first term and common difference of A.P.
5(a+4a) = 8(a + 7d) $(a_n = a+(n-1)d)$
5a + 20d = 8a +56d
5a - 8a = 56d-20d
-3a = 36d
a = -12d....(1)
$a_{13}$ = a+12d = -12d+12d = 0 (using 1)
∴$a_{13}$ = 0