# What is the sum of the geometric sequence 2, 10, 50, … if there are 8 terms?

The formula for the sum to n terms of such a geometric series is given by :

So in this particular case it becomes

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The sum of a geometric sequence can be calculated using the formula:

[ S_n = a \frac{{r^n - 1}}{{r - 1}} ]

Where:

- ( S_n ) 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

Given the sequence 2, 10, 50, ..., we can observe that:

- ( a = 2 ) (the first term)
- ( r = 5 ) (each term is obtained by multiplying the previous term by 5)

Now, we can plug these values into the formula:

[ S_8 = 2 \frac{{5^8 - 1}}{{5 - 1}} ]

[ S_8 = 2 \frac{{390625 - 1}}{{4}} ]

[ S_8 = 2 \frac{{390624}}{{4}} ]

[ S_8 = 2 \times 97656 ]

[ S_8 = 195312 ]

So, the sum of the geometric sequence 2, 10, 50, ... with 8 terms is 195312.

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