What is the sum of the first eight terms in this series?
$2+10+50+250+\cdots$
The given series is \(2 + 10 + 50 + 250 + \cdots\). By observing the terms, we see that each term is obtained by multiplying the previous term by \(5\). Specifically, \(10 = 2 \times 5\), \(50 = 10 \times 5\), and \(250 = 50 \times 5\). This is a geometric series where the first term \(a = 2\) and the common ratio \(r = 5\). We are asked to find the sum of the first \(n = 8\) terms of this series.
The formula for the sum of the first \(n\) terms of a geometric series is given by:
\[S_n = a \frac{r^n - 1}{r - 1}\]
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. For our specific case, \(a = 2\), \(r = 5\), and \(n = 8\). Substituting these values into the formula, we have:
\[S_8 = 2 \times \frac{5^8 - 1}{5 - 1}\]
Now, we simplify the expression:
\[S_8 = 2 \times \frac{5^8 - 1}{4} = \frac{5^8 - 1}{2}\]
First, calculating \(5^8\):
\(5^1 = 5\)
\(5^2 = 25\)
\(5^3 = 125\)
\(5^4 = 625\)
\(5^5 = 3,125\)
\(5^6 = 15,625\)
\(5^7 = 78,125\)
\(5^8 = 390,625\)
Then, substituting this value back into the expression:
\[S_8 = \frac{390,625 - 1}{2} = \frac{390,624}{2} = 195,312\]
Comparing this result to the given options, we find that the correct sum is \(195,312\).
Geometric series are essential in various fields, such as finance and population studies. For instance, if an investment grows by a fixed percentage each year, calculating the total value over a period is essentially finding the sum of a geometric series. If you invest \(\$2\) and it grows by 400% (so each year it becomes 5 times the previous value), you can predict the total growth over 8 years using the geometric sum formula, helping you plan for future financial goals.