UPSC NDA Math Model Paper 3

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

179

321

.