Let A1 and A2 be the two series with first terms a1 and a2 and the common differences d1 and d2 respectively. Using the formula for the sum of n terms, we can say that: S1n=
n
2
[2a1+(n−1)d1] S2n=
n
2
[2a2+(n−1)d2] According to the question:
S1n
S2n
=
5n+4
9n+6
⇒
2a1+(n−1)d1
2a2+(n−1)d2
=
5n+4
9n+6
⇒
a1+
(n−1)
2
d1
a2+
(n−1)
2
d2
=
5n+4
9n+6
........ (1) In order to find the ratio of the 18th terms, we must have:
n−1
2
=17 ⇒n−1=34 ⇒n=35 Substituting n = 35 in the equation (1) above, we get: ⇒
a1+17d1
a2+17d2
=
5×35+4
9×35+6
⇒
a1+17d1
a2+17d2
=
175+4
315+6
⇒
a1+17d1
a2+17d2
=
179
321
Therefore, the ratio of the 18th terms of both the AP is