Evaluate the Summation sum from k=1 to 5 of 3(1/5)^(k-1)
The question is asking for the evaluation of a finite geometric series. The specific series in question starts at k=1 and goes up to k=5. For each term in the series, you should multiply 3 by the term (1/5) raised to the power of (k-1). You are supposed to calculate the sum of the first five terms of this series.
$\sum_{k = 1}^{5} 3 \left(\left(\right. \frac{1}{5} \left.\right)\right)^{k - 1}$
Answer
Solution:
Step:1
Write out the terms of the summation for each value of \( k \) from 1 to 5.
\[ 3 \left( \frac{1}{5} \right)^{0} + 3 \left( \frac{1}{5} \right)^{1} + 3 \left( \frac{1}{5} \right)^{2} + 3 \left( \frac{1}{5} \right)^{3} + 3 \left( \frac{1}{5} \right)^{4} \]
Step:2
Calculate the sum of the terms.
\[ \frac{2343}{625} \]
Step:3
Express the summation result in various formats.
Exact Form: \( \frac{2343}{625} \) Decimal Form: \( 3.7488 \) Mixed Number Form: \( 3 \frac{468}{625} \)
Solution:"The summation of the series for \( k \) ranging from 1 to 5 is calculated by expanding the series, simplifying the terms, and then summing them up to get the final result in various forms including exact, decimal, and mixed number formats."
Knowledge Notes:
The problem involves evaluating a finite geometric series. A geometric series is a series with a constant ratio between successive terms. The general form of a geometric series is \( a + ar + ar^2 + \ldots + ar^{n-1} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
In this problem, the first term \( a \) is \( 3 \), and the common ratio \( r \) is \( \frac{1}{5} \). The series is finite because it only sums the first five terms. The \( k \)-th term of the series is given by \( 3 \left( \frac{1}{5} \right)^{k-1} \).
To evaluate the summation, we expand the series by plugging in the values of \( k \) from 1 to 5, then simplify each term, and finally, sum them up to get the total. The result can be expressed in different forms, such as an exact fraction, a decimal approximation, or a mixed number, which is a combination of a whole number and a proper fraction.
