What is the sum of the geometric sequence: 2, 10, 50 when there are 8 terms?

The given geometric sequence starts with the first term a = 2, and the common ratio r can be found by dividing the second term by the first term. Thus, the common ratio is:

r = 10 / 2 = 5

Now that we have identified the first term and the common ratio, we can use the formula for the sum of a geometric series:

S_n = a * (1 – rⁿ) / (1 – r)

where:

• S_n is the sum of the first n terms.
• a is the first term.
• r is the common ratio.
• n is the number of terms.

In this case, we have:

• a = 2
• r = 5
• n = 8

Substituting these values into the sum formula:

S₈ = 2 * (1 – 5⁸) / (1 – 5)

Calculating 5⁸:
5⁸ = 390625

Then:

S₈ = 2 * (1 – 390625) / (1 – 5)

S₈ = 2 * (-390624) / (-4)

S₈ = 2 * 97656

S₈ = 195312

Thus, the sum of the first 8 terms of the geometric sequence 2, 10, 50 is 195312.