Question

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

Answer 1

Here is the solution of the problem if the sum of 9 terms of A.P is 162.
Given,
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 first term of A.P is 6
The 15th term of A.P is 48