The given geometric sequence starts with the terms 2, 10, and 50. To find the sum of this sequence over 8 terms, we first need to identify the common ratio and the formula for the sum of a geometric series.
In a geometric sequence, each term can be found by multiplying the previous term by a constant called the common ratio. Let’s find the common ratio (r) between the first two terms:
r = second term / first term = 10 / 2 = 5
Next, we can verify this ratio with the subsequent terms:
r = third term / second term = 50 / 10 = 5
So, the common ratio is indeed 5.
Now, we can see that the formula for the sum (S) of the first n terms of a geometric sequence is:
Sn = a * (1 – rn) / (1 – r)
Where:
- Sn is the sum of the first n terms
- a is the first term
- r is the common ratio
- n is the number of terms
Substituting the known values into the formula:
- a = 2
- r = 5
- n = 8
We can now calculate:
S8 = 2 * (1 – 58) / (1 – 5)
Calculating 58 gives us 390625:
S8 = 2 * (1 – 390625) / (1 – 5) = 2 * (-390624) / (-4)
This simplifies to:
S8 = 2 * 97656 = 195312
Therefore, the sum of the first 8 terms of the geometric sequence is 195312.