9–15. Geometric sums Evaluate each geometric sum.

Question

{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

Accepted Answer

Welcome back, everyone. Find the sum sigma from J equals 0 up to 10 of 5 divided by 8. Raise the power of 2 J. Round your answer to two decimal places. A 1.32, B 1.64, C 2.12, and D 2.64. For this problem, let's show that our series is a geometric series. We have sigma from J equals 0 up to 10 of 5 divided by 8. Raise the power of 2J. Now the main problem is the exponent. It's not J, it's 2J. So what we can do to clearly identify our common ratio for the geometric series is simply apply the properties of exponents. We can write this expression as 5 divided by 8 squared and then raise the power of J, right. So we're going to get sigma from J equals 0 up to 10 of 25 divided by 64, raise the power of J. And then we can clearly see what the common ratio R is, right? R is equal to 25 divided by 64. We can also identify our first term, taking 25 divided by 64. And raising it to the power of 0 because our first term begins with J equals 0. We got one. And now we're going to identify the sum as N, right? Let's remember that SN, the sum of geometric series, is equal to A1, multiplied by. 1 minus R raises the power of N divided by 1 minus R. A1 is our first term, so we take 1. Multiply by 1 minus R, raise the power of N. We already know that R, which is our common ratio, is 25 divided by 64. Now N is the number of terms and we can identify N as. 10 minus 0. Plus 1, that's 11, right? So we're basically evaluating. As of 11, because we have a total of 11 terms, meaning we can say. S 11 equals 1 multiplied by 1 minus 25 divided by 64, raise to the power of. 11 Divided by 1 minus R, right? So, we basically take 1 minus 25 divided by 64. Since we are rounding in this problem, we can basically use a calculator to perform our calculation, and we're going to get approximately 1.64 rounded to two decimal places, which corresponds to the answer choice B. Thank you for watching.

9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

Key Concepts

Geometric Series

Sum Formula for a Finite Geometric Series

Evaluating Powers and Substitution

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