Evaluate the Summation sum from i=1 to 5 of 3*2^i
The question asks for the calculation of a summation, specifically, the sum of terms that follow a defined rule with i as the variable index. The index i begins at 1 and continues through to 5, incrementing by 1 with each step. For each value of i, the term to be added to the sum is 3 multiplied by 2 raised to the power of i. The question requires the evaluation of this sum by plugging in each integer value of i from 1 to 5 into the term, calculating the result, and then adding all these results together to find the final sum.
$\sum_{i = 1}^{5} 3 \cdot 2^{i}$
Answer
Solution:
Step 1: Expansion of the Series
Write out the terms of the series by substituting $i$ with each integer from 1 to 5.
$3 \cdot 2^{1} + 3 \cdot 2^{2} + 3 \cdot 2^{3} + 3 \cdot 2^{4} + 3 \cdot 2^{5}$
Step 2: Calculation of the Sum
Compute the sum of the terms in the expanded series.
$3(2 + 4 + 8 + 16 + 32) = 3 \cdot 62 = 186$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a common mathematical operation used to add together a sequence of terms generated by a formula. The specific summation in question is of the form $\sum_{i=1}^{n} a \cdot r^{i}$, where $a$ is a constant, $r$ is the base of the exponential, and $i$ is the index of summation.
Relevant knowledge points include:
Geometric Series: A series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). In this case, the series is not a perfect geometric series because of the additional multiplication by 3.
Summation Notation: The sigma notation $\sum$ represents the sum of a sequence of numbers. The expression under the sigma indicates the general term of the series, and the limits of summation are given above and below the sigma.
Exponents: The notation $b^{n}$, where $b$ is the base and $n$ is the exponent, represents the operation of multiplying the base $b$ by itself $n$ times.
Arithmetic Operations: The process involves basic arithmetic operations such as multiplication and addition.
In this problem, the summation is evaluated by expanding the series to show each term and then simplifying the expression by performing the arithmetic operations to find the sum.
