Question: Which of the following is one way the expression \[ \sum_{n=18}^{64}…

Which of the following is one way the expression

\[ \sum_{n=18}^{64} \left(\frac{1}{2^n}\right) \]

can be rewritten using summation properties?

1. \(\frac{1}{2} \left(\frac{64(65)}{2}\right) - \frac{1}{2} \left(\frac{19(20)}{2}\right)\)
2. \(\frac{1}{2} \left(\frac{64(65)}{2}\right)\)
3. \(\frac{1}{2} \left(\frac{64(65)}{2}\right) - \frac{1}{2} \left(\frac{17(18)}{2}\right)\)
4. \(\frac{1}{2} \left(\frac{46(47)}{2}\right)\)

Solution

To rewrite the expression using summation properties, we need to identify the correct manipulation of the original expression: \[ \sum_{n=18}^{64} \left( \frac{1}{2} \right)^n \] We can recognize this as a geometric series with the first term \( a = \left(\frac{1}{2}\right)^{18} \) and the common ratio \( r = \frac{1}{2} \). The sum of a finite geometric series is given by: \[ S = a \frac{1 - r^N}{1 - r} \] where \(N\) is the number of terms. To simplify the summation: 1. The series starts at \(n = 18\) and ends at \(n = 64\), so the number of terms \(N = 64 - 18 + 1 = 47\). 2. Using the properties of sums, we can rewrite the expression as: \[ \sum_{n=0}^{64} \left( \frac{1}{2} \right)^n - \sum_{n=0}^{17} \left( \frac{1}{2} \right)^n \] Using these properties, we look for an option that reflects the subtraction of two geometric series. The correct choice is: \[ \frac{1}{2} \binom{64}{65} - \frac{1}{2} \binom{17}{18} \] This option correctly represents the original series as a difference of two series, each calculated from \(n=0\) to different limits.