What is the sum of the geometric series 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + … + 2^9? a)1,023 b)511 c)2,407 d)1,012
Mathematics · College · Thu Feb 04 2021
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To find the sum of the geometric series 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + … + 2^9, we can use the formula for the sum of a finite geometric series:
The sum S of a geometric series with the first term a, common ratio r, and n terms is given by:
\[ S = a \times \frac{1 - r^n}{1 - r}, \text{ where } r \neq 1 \]
In this case, our first term, a, is 2^0 which equals 1, our common ratio r is 2 (because we are doubling each term) and we have n = 10 terms (from 2^0 to 2^9).
Applying the formula:
\[ S = 1 \times \frac{1 - 2^{10}}{1 - 2} \]
Now, calculate the exponent:
\[ 2^{10} = 1024 \]
And substitute it in:
\[ S = \frac{1 - 1024}{1 - 2} \] \[ S = \frac{-1023}{-1} \] \[ S = 1023 \]
So the sum of the series is 1023, which corresponds to option a) 1,023.