Question

The sum of the first q terms of an ap is 162. The ratio of its 6th term to its 13th term is 1:2. Find the 1st and 15th term of the ap

Answer 1

Sum of terms= 162

Ratio of 6th term and 13th term= 1:2

Formula is an= a1+(n-1)d

Then the sixth term a6= a1+5d

The 13th term is a13=a1+12d

As given a6/a13 is 1/2

(a1+5d)/(a1+12d)=1/2

After cross multiplication

2(a1+5d)=1(a1+12d)

2a1+10d=a1+12d

2a1-a1=12d-10d

Therefore the first term a1=2d

Sn=(n/2) [2a+(n-1)d]

S9=(9/2) [2(2d)+(9-1)d]

S9=(9/2)(4d+8d)

162=(9/2)(12d)

162=9(6d)

54d=162

d=3

a1=2d=2(3)=6

a15=6+(15-1)3=6+3(14)=48

The 1st term of A.P is 6.

The 15th term of A.P is 48.