Question

Write the geometric series for the following sequence that sums the first 50 terms in sigma notation: 5, -5/2, 5/4, ...

Answer 1

To express the given sequence \( 5, -\frac{5}{2}, \frac{5}{4}, \ldots \) as a geometric series in sigma notation, we first identify the first term and the common ratio.

1. **First Term (\(a\))**: The first term of the sequence is \(5\).

2. **Common Ratio (\(r\))**: We compute the common ratio by dividing the second term by the first term:
\[
r = \frac{-\frac{5}{2}}{5} = -\frac{1}{2}
\]
We can verify this with the next term as well:
\[
r = \frac{\frac{5}{4}}{-\frac{5}{2}} = \frac{5}{4} \cdot \frac{-2}{5} = -\frac{1}{2}
\]
Thus, the common ratio is confirmed to be \( -\frac{1}{2} \).

The general form for a geometric series can be expressed as:
\[
a + ar + ar^2 + ar^3 + \ldots + ar^{n-1}
\]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

For our sequence, we have:

- \(a = 5\)
- \(r = -\frac{1}{2}\)
- We want to sum the first 50 terms, so \(n = 50\).

The sum of the first \(n\) terms can be written in sigma notation as:
\[
\sum_{k=0}^{49} 5 \left(-\frac{1}{2}\right)^k
\]

Therefore, the geometric series for the given sequence that sums the first 50 terms in sigma notation is:
\[
\sum_{k=0}^{49} 5 \left(-\frac{1}{2}\right)^k
\]