Evaluate the Summation sum from i=1 to 5 of (1/5)^i
This problem is asking for the calculation of a finite series. Specifically, it wants you to calculate the sum of the series where each term is the reciprocal of five (1/5) raised to the power of the current term's position in the series, starting from the position i=1 and continuing until i=5. In other words, you need to find the sum of the first five terms of this geometric sequence.
$\sum_{i = 1}^{5} \left(\left(\right. \frac{1}{5} \left.\right)\right)^{i}$
Answer
Solution:
Step 1: Write out the terms of the series.
Calculate each term by raising $\frac{1}{5}$ to the power of $i$, where $i$ ranges from 1 to 5.
$\left(\frac{1}{5}\right)^1 + \left(\frac{1}{5}\right)^2 + \left(\frac{1}{5}\right)^3 + \left(\frac{1}{5}\right)^4 + \left(\frac{1}{5}\right)^5$
Step 2: Perform the calculations for each term.
Combine the terms to find the sum of the series.
$\frac{781}{3125}$
Step 3: Express the sum in various formats.
The sum can be represented in different ways.
Exact Form: $\frac{781}{3125}$ Decimal Form: $0.24992$
Solution: "The summation of the series from i=1 to 5 of $(1/5)^i$ is $\frac{781}{3125}$, which can also be expressed as $0.24992$ in decimal form."
Knowledge Notes:
The problem involves evaluating a finite geometric series. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this case, the common ratio is $\frac{1}{5}$.
To solve the problem, we:
Expanded the series by computing each term individually.
Summed the terms to find the total.
Expressed the result in both exact (fractional) form and approximate (decimal) form.
This problem demonstrates the concept of a geometric series and how to evaluate its sum when the number of terms and the common ratio are known. It also shows how to express the result in different forms, which is useful in various mathematical and real-world applications.
