Problem 36 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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 \(S = \frac{(\frac{1}{9})(1 - (\frac{1}{3})^6)}{1 - \frac{1}{3}}\).

Step 3: 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.