What is the 8th term of the geometric sequence where a1 is 256 and a3 is 16?

To find the 8th term of the geometric sequence, we first need to establish the common ratio of the sequence.

We know:

• The first term, a₁, is 256.
• The third term, a₃, is 16.

In a geometric sequence, each term can be expressed as:

a_n = a₁ × r^(n-1)

Where r is the common ratio. Using this formula, we can express the third term as:

a₃ = a₁ × r^(3-1) = a₁ × r²

Substituting in the known values:

16 = 256 × r²

To find r², we can rearrange this equation:

r² = 16 / 256

Calculating this gives:

r² = 1 / 16

Taking the square root of both sides, we find:

r = 1/4

With the common ratio known, we can now find the 8th term:

a₈ = a₁ × r^(8-1)

Substituting the values we have:

a₈ = 256 × (1/4)⁷

Calculating (1/4)⁷ = 1/16384, we continue:

a₈ = 256 / 16384

This simplifies to:

a₈ = 1/64

Thus, the 8th term of the geometric sequence is 1/64.