9–15. Geometric sums Evaluate each geometric sum.

Question

{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

Accepted Answer

Welcome back, everyone. Find the sum sigma from J equals 0 up to 7 of -13 raise to the power of J. Round your answer to two decimal places. A 0.75, B 0.90, C-1.12, and D 1.50. For this problem, let's recognize that our series sigma from J equals 0 up to 7 of 1/3 raises the power of J corresponds to a geometric series in the form of sigma from J equals 0 up to 7 of R raise the power of J, where R is the common ratio. So we can define our common ratio R as 1/3. Let's recall that we can compute the sum of the geometric series using the formula S N equals A, where A is the first term of the series multiplied by 1 minus R raises the power of N. Divided by 1 minus r. So, we already know what R is, it's 1/3. We're going to compute our first term, which is A. Taking the general term formula and that's negative 1/3 raises the power of J, and because J begins with zero, we basically have to raise 1/3 to the power of 0, which is going to be equal to 1. Well then, so we have our first term A, which is 1. We're multiplying 1 by 1 minus R, which is 1/3. Race to the power of N. So basically we want to identify N, right? How many terms do we have? Because J begins with 0 and ends with 7, we have to find the difference between 7 and 0 and add 1, which gives us a total of 8 terms. So now using the formula we can show that S8 because now N is equal to 8th. Is equal to. 1 multiplied by 1 minus 1/3 raises the power of eight. Divided by 1 minus R, which is 1 minus 1/3. We can simplify slightly to show that this is 1 minus 1/3 raises the power of 8th, right, because we're raising to an even exponent. Divided by 1 minus 1/3 is 1 + 1/3, which is 4/3. And now we can calculate the result. Specifically, we can replace division by 4/3 with multiplication by 3/4. So, 3/4 is going to be multiplied by 1 minus 1/3 raised to the power of 8th. Approximating the answer to two decimal places, we get 0.75. Which corresponds to the answer choice a. Thank you for watching.

9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

Key Concepts

Geometric Series

Sum Formula for Finite Geometric Series

Use of Technology in Calculations

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