What is the 7th term of the geometric sequence where the first term (a1) is 256 and the third term (a3) is 16?

To find the 7th term of a geometric sequence, we first need to identify the common ratio and use it to determine the 7th term.

In a geometric sequence, each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio (often denoted as r). The general formula for the nth term (a_n) is given by:

a_n = a₁ imes r^(n-1)

Given:

• a₁ = 256
• a₃ = 16

We can write the formula for the third term:

a₃ = a₁ imes r²

Plugging in the known values:

16 = 256 imes r²

To isolate r², divide both sides by 256:

r² = rac{16}{256}

Calculating the right side gives:

r² = rac{1}{16}

Taking the square root of both sides, we find:

r = rac{1}{4}

Now that we have the common ratio, we can find the 7th term:

a₇ = a₁ imes r^(7-1) = 256 imes r⁶

Calculating r⁶:

r⁶ = igg(rac{1}{4}igg) = rac{1}{4096}

Now substitute back into the formula:

a₇ = 256 imes rac{1}{4096}

Calculating this gives:

a₇ = rac{256}{4096} = rac{1}{16}

Thus, the 7th term of the geometric sequence is:

a₇ = rac{1}{16}