Question

The sum of third and seventh term of an A. P. is 6 and their product is 8. Find the first term and the common difference of the A. P.

Answer 1

t_n = a + (n - 1)d

∴ t₃ = a + (3 - 1)d = a + 2d

t₇ = a + (7 - 1)d = a + 6d

 t₃ + t₇ = (a + 2d) + (a + 6d) = 2a + 8d

∴ 2a + 8d = 6

a + 4d = 3 ..........(I)

t₃ × t ₇ = (a + 2d) (a + 6d)

= (a + 4d - 2d) (a + 4d + 2d)

= (3 - 2d) (3 + 2d) .......... from (I)

∴ (3 - 2d) (3 + 2d) = 8 

∴ 9 - 4d² = 8 

∴ 4d² = 1 d² = 1 / 4  d = 1 / 2 or d = - 1 / 2

Or

∴ the first term of the A. P. is 1 and the common difference is 1/2.

or, the first term of the A. P. is 5 and the common difference is - 1 / 2 .