A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. In your case, the sequence begins with the terms 4, 16, and 64. Let’s find the common ratio by dividing the second term by the first term:
Common Ratio (r):
- r = 16 / 4 = 4
- r = 64 / 16 = 4
We can see that the common ratio is 4.
The first term of the sequence (a) is 4 and the common ratio (r) is 4. To find the sum of the first n terms of a geometric sequence, we can use the formula:
Sn = a * (1 – rn) / (1 – r)
Where:
- Sn is the sum of the first n terms
- a is the first term of the sequence
- r is the common ratio
- n is the number of terms
In your case, we need to find the sum of the first 8 terms (n = 8):
S8 = 4 * (1 – 48) / (1 – 4)
Calculating this:
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 given geometric sequence is 87380.