Question

In a geometric sequence, a₆ = 8 and a₇ = 32. Use the geometric mean to find the value of a₂₆.

(A) 4
(B) 16
(C) 64
(D) 128

Answer 1

The value of a₂₆ in the geometric sequence is 2^45, which is equivalent to 2,147,483,648.

Step-by-step explanation:

To find the value of a₂₆ in a geometric sequence, we can use the formula for the nth term:

aₙ = a₁ * r^(n-1)

where aₙ is the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term we want to find.

Given that a₆ = 8 and a₇ = 32, we can set up the following equations:

8 = a₁ * r^(6-1)

32 = a₁ * r^(7-1)

Dividing these two equations, we get:

32/8 = r^(7-1) / r^(6-1)

Simplifying, we have:

4 = r^(7-6)

4 = r

Now we can find a₂₆:

a₂₆ = a₁ * r^(26-1) = 8 * 4^(26-1) = 8 * 4^25

Simplifying, we get:

a₂₆ = 8 * 2^50 = 16 * 2^49 = 32 * 2^48 = 64 * 2^47 = 128 * 2^46 = 256 * 2^45 = ... = 2^45 * 2^45 = (2^45)^2

Therefore, the value of a₂₆ is 2^45, which is equivalent to 2,147,483,648.