Question

6. The sum of the 3rd and the 7th terms of an
A.P. is 6 and their product is 8. Find the first
term and the common difference.​

Answer 1

ANSWER:-

Given:

The sum of the 3rd and the 7th term of an A.P. is 6 and their product is 8.

To find:

Find the first term & the common difference.

Solution:

Let a & d be the first term and common difference of A.P.

nth term of A.P.,

${}^{a} n = a + (n - 1)d$

Therefore,

a3= a+ (3-1)d

=) a + 2d

a7 = a+ (7-1)d

=) a + 6d

According to this question:

a3 + a7= 6

=) (a+2d) + (a+6d)= 6

=) a+2d + a+6d = 6

=) 2a + 8d = 6

=) a + 4d = 3

=) a= 3 -4d..............(1)

Therefore,

Their product is 8.

a3 × a7 = 8

=) (a+ 2d)× (a+6d)= 8

=) (3-4d + 2d)× (3-4d +6d)=8. [Using(1)]

=) (3- 2d) (3+2d)= 8

=) 9 -4d² = 8

=) 4d² = 9-8

=) 4d² = 1

=) d² = 1/4

=) d = √1/4

=) d = ±1/2

Now,

When d = 1/2

a= 3- 4d

=) a = 3- 4× 1/2

=) a= 3 - 2

=) a = 1

When, d = -1/2

a= 3 -4d

=) a = 3 -4×-1/2

=) a = 3 - (-2)

=) a = 3 +2

=) a= 5

Hence,

First term,(a) = 1 & 5

Common difference,(d)= ±1/2

Hope it helps ☺️