Question

The 4th and 7th terms of a G.P. are 16 and 2 respectively.
Find:
a. The common ratio
b. The first term

Answer 1

a. From the formula for the nth term of a G.P., we know that:

𝑎_𝑛 = 𝑎_1 × 𝑟^(𝑛−1)

So we can write:

𝑎_4 = 𝑎_1 × 𝑟^(4−1) = 16
𝑎_7 = 𝑎_1 × 𝑟^(7−1) = 2

Dividing the second equation by the first, we get:

(𝑎_1 × 𝑟^6) / (𝑎_1 × 𝑟^3) = 2/16

Simplifying, we get:

𝑟^3 = 1/4

Taking the cube root of both sides, we get:

𝑟 = 1/2

Therefore, the common ratio is 1/2.

b. We can use the first equation above to solve for 𝑎_1:

𝑎_1 = 16 / 𝑟^3

Substituting the value we found for 𝑟, we get:

𝑎_1 = 16 / (1/8) = 128

Therefore, the first term is 128.