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Chapter 8: Problem 36

Find the indicated sum. Use the formula for the sum of the first \(n\) terms of a geometric sequence. $$\sum_{i=1}^{6}\left(\frac{1}{3}\right)^{i+1}$$

Short Answer

Expert verified
The sum of the first six terms of the defined geometric sequence is given by \(S = \frac{(\frac{1}{9})(1 - (\frac{1}{3})^6)}{1 - \frac{1}{3}}\). One must compute the operations on the right side of this equation to obtain the numeric solution.

Step by step solution

01

Identify the parameters of the geometric sequence

The first term \(a\) of the sequence is obtained by replacing \(i\) with 1 in the function, yielding \((\frac{1}{3})^{1+1} = \frac{1}{9}\). The common ratio \(r\) is \(\frac{1}{3}\) because this is the factor that is multiplied by each term to get to the next term in the sequence. The number of terms \(n\) is given as 6.
02

Apply the sum formula

Now substitute the identified \(a\), \(r\) and \(n\) into the formula for the sum of the first \(n\) terms of a geometric sequence, \(S = \frac{a(1 - r^n)}{1 - r}\). This gives us \(S = \frac{(\frac{1}{9})(1 - (\frac{1}{3})^6)}{1 - \frac{1}{3}}\).
03

Simplify the expression

Multiply out the terms in the numerator and simplify the expression by performing the operations. After doing so, you receive the solution for this exercise.

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