Geometric sums
Evaluate the geometric sums
∑ (from...

Question

Geometric sums

Evaluate the geometric sums

∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

Accepted Answer

Welcome back, everyone. Consider the geometric sums A equals sigma from K equals 0 to 8 of 0.5 to the power of K and B equals sigma from k equals 4/18 of 0.5 K. Evaluate both sums and round your answers to three decimal places. For this problem, we're dealing with geometric series and we have to recall that one of the formulas that allows us to calculate the sum of the geometric series from an initial index to some final index would be a 1 minus A. N + 1 divided by 1 minus R. So now E1 is the first term, EN plus 1 is the term after the last term, N. And R is the common ratio. Let's notice that for each geometric sums, the common ratio R is equal to 0.5. That's the part that contains the exponent, right? And we have our initial index and our final index. So we can evaluate A using the formula. A1 is going to be our first term since the initial index is K equals 0, we get 0.5 raise to the power of 0, and we're going to subtract A N + 1. So that'd be 0.5 raises the power of A + 1, right? Specifically, this is because our final term has. N of 8, right? So N + 1 would be 8 + 1. And we're dividing by 1 minus the common ratio R. The common ratio is 0.5. We can simplify and show that this is equal to 1 minus 0.5 to the power of 9. Divided by 0.5. And using a calculator, we can approximate the results, that would be approximately 1.996. So, this is our first answer. Now to evaluate the B, we're going to use the same formula. Now, the word first term is going to be at K equals 4. This is our initial index. So we get 0.5, raise to the power of 4 minus 0.5, raise to the power of 8 plus 1. Once again, our final index is 8, so, N equals 8. And we're dividing by 1 minus 0.5. The common ratio is the same. Simplifying, we get 0.5, raise to the power of 4 minus 0.5, raise to the power of 9. Divided by 0.5. We simply, 0.5 raises the power of 3 minus 0.5, raise the power of eight, right, using the properties of fractions and exponents. And then we can approximate the results. This is going to be 0. 121 rounded to 3 decimal places. Thank you for watching.

Geometric sums
Evaluate the geometric sums
∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

Key Concepts

Geometric Series

Formula for the Sum of a Finite Geometric Series

Index Shifting in Summations

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