How do you find the 16th term of a geometric sequence if the first term is 4 and the eighth term is 8748?

To find the 16th term of a geometric sequence where the first term (a₁) is 4 and the eighth term (a₈) is 8748, we first need to determine the common ratio (r) of the sequence.

A geometric sequence is defined by the formula for the nth term:

a_n = a₁ imes r^(n-1),

where:

• a_n = the nth term of the sequence
• a₁ = the first term
• r = common ratio
• n = term number

We know:

• a₁ = 4
• a₈ = 8748

Using the formula to express a₈:
a₈ = a₁ imes r^(8-1).
This simplifies to:

8748 = 4 imes r⁷

Next, divide both sides by 4 to isolate r⁷:

r⁷ = 8748 / 4 = 2187

Now, we need to find r. Taking the seventh root of both sides gives us:

r = 2187^(1/7)

Calculating the seventh root:

r = 3

Now that we have the common ratio, we can find the 16th term (a₁₆):

a₁₆ = a₁ imes r^(16-1), which simplifies to:

a₁₆ = 4 imes 3¹⁵

First, we calculate 3¹⁵:

3¹⁵ = 14348907

Then, multiplying this by 4:

a₁₆ = 4 imes 14348907 = 57395628

Thus, the 16th term of the geometric sequence is 57395628.