What is the sum of the geometric sequence 4, 16, 64 if there are 8 terms?

To find the sum of the first 8 terms of the geometric sequence 4, 16, 64, we first need to identify the common ratio and then apply the formula for the sum of a geometric series.

The first term (a) in this sequence is 4. To find the common ratio (r), we can divide the second term by the first term:

r = 16 / 4 = 4

Now, we have:

• First term (a) = 4
• Common ratio (r) = 4
• Number of terms (n) = 8

The formula for the sum of the first n terms of a geometric series is:

S_n = a * (1 – rⁿ) / (1 – r)

Substituting the values we have:

S₈ = 4 * (1 – 4⁸) / (1 – 4)

Now, calculate 4⁸:

4⁸ = 65536

Then, plug it back into the formula:

S₈ = 4 * (1 – 65536) / (1 – 4)

S₈ = 4 * (-65535) / (-3)

S₈ = 4 * 21845

S₈ = 87380

Therefore, the sum of the first 8 terms of the geometric sequence 4, 16, 64 is 87380.