To write the sigma notation for the given finite geometric sequence, we first identify the common ratio and the number of terms.
The sequence provided is:
8, 40, 200, 1000, 5000
-
Identify the first term (a): The first term \(a\) = 8.
-
Identify the common ratio (r): To find the common ratio, we can divide the second term by the first term: \[ r = \frac{40}{8} = 5 \] We can confirm this with the other terms:
- \( \frac{200}{40} = 5 \)
- \( \frac{1000}{200} = 5 \)
- \( \frac{5000}{1000} = 5 \)
So, the common ratio \(r\) = 5.
- Count the number of terms (n): The number of terms in the sequence is 5.
Using the formula for the \(n\)-th term of a geometric sequence: \[ a_n = a \cdot r^{n-1} \]
The sum of the first \(n\) terms of a geometric series can be expressed in sigma notation as: \[ \sum_{k=0}^{n-1} a \cdot r^k \]
Now, substituting the values we have:
- \(a = 8\)
- \(r = 5\)
- \(n = 5\)
The sigma notation will be: \[ \sum_{k=0}^{4} 8 \cdot 5^k \]
So, the final answer in sigma notation is:
\[
\sum_{k=0}^{4} 8 \cdot 5^k
\]