A GP has a first term of a, a common ratio of r and its 6th term is 768. Another GP has a first term of a, a common ratio of 6r and its 3rd term is 3456. Evaluate a and r.

Answer 1

In GP:

an = a ∙ rⁿ⁻¹

First condition:

a6 = 768

a ∙ r⁵ = 768

Second condition:

a3 = 3456

a ∙ ( 6 r)² = 3456

a ∙ 36 r² = 3456

a ∙ r⁵ = 768
:
a ∙ 36 r² = 3456
_____________

a / a ∙ r⁵ / 36 r² = 768 / 3456

1 ∙ r³ / 36 = 384 ∙ 2 / 384 ∙ 9

r³ / 36 = 2 / 9

Multiply both sides by 36

r³ = 2 ∙ 36 / 9

r³ = 72 / 9

r³ = 8

r = ³√ 8

r = 2

First condition:

a6 = 768

a ∙ r⁵ = 768

a ∙ 2⁵ = 768

a ∙ 32 = 768

a = 768 / 32

a = 24

Answer 2

r=2, a=24