The first term if an Arithmetic Progression is -8. The ratio of the 7th term to the 9th term is 5:8. Calculate the common difference of the progression.

Answer 1

For an arithmetic progression, a_n = a₁ + (n-1)d

Thus, the seventh term is a₁ + (7-1)d = a₁ + 6d, and the ninth term is a₁ + (9-1)d = a₁ + 8d

Then, if their ratio is 5:8,

(a₁ + 6d)/(a₁ + 8d) = 5/8

As the first term, a₁, equals -8

(-8 + 6d)/(-8 + 8d) = 5/8

Cross-multiplying,

(-8 + 6d)8 = (-8 + 8d)5

-64 + 48d = -40 + 40d

8d = 24

d = 3

To check, a₇ = -8 + 6(3) = 10; a₉ = -8 + 8(3) = 16

10/16 = 5/8