Evaluate the Summation sum from 0 to 3 of 2^(n+1)
The problem presents a mathematical task requiring you to calculate the sum of a series. Specifically, the series in question is the summation of terms of the form 2^(n+1), where the exponent is one more than the current term number n. The sum should be evaluated for all integers n ranging from the initial term (where n equals 0) to the final term (where n equals 3). Essentially, you are asked to add together the values of this expression when n is each of the numbers 0, 1, 2, and 3.
$\sum_{0}^{3} 2^{n + 1}$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer value of $n$ from $0$ to $3$: $2^{0 + 1} + 2^{1 + 1} + 2^{2 + 1} + 2^{3 + 1}$.
Step 2:
Proceed to simplify the expression.
Step 2.1:
Combine the base and exponent where $n = 0$: $2^{1} + 2^{1 + 1} + 2^{2 + 1} + 2^{3 + 1}$.
Step 2.2:
Calculate $2$ to the first power: $2 + 2^{1 + 1} + 2^{2 + 1} + 2^{3 + 1}$.
Step 2.3:
Combine the base and exponent where $n = 1$: $2 + 2^{2} + 2^{2 + 1} + 2^{3 + 1}$.
Step 2.4:
Calculate $2$ to the second power: $2 + 4 + 2^{2 + 1} + 2^{3 + 1}$.
Step 2.5:
Add the first two terms: $6 + 2^{2 + 1} + 2^{3 + 1}$.
Step 2.6:
Combine the base and exponent where $n = 2$: $6 + 2^{3} + 2^{3 + 1}$.
Step 2.7:
Calculate $2$ to the third power: $6 + 8 + 2^{3 + 1}$.
Step 2.8:
Add the results of the first three terms: $14 + 2^{3 + 1}$.
Step 2.9:
Combine the base and exponent where $n = 3$: $14 + 2^{4}$.
Step 2.10:
Calculate $2$ to the fourth power: $14 + 16$.
Step 2.11:
Add the final two terms to get the sum: $30$.
Knowledge Notes:
The problem involves evaluating a finite series, specifically a geometric series where the common ratio is a power of $2$. The series is defined as the sum of terms $2^{n+1}$ where $n$ ranges from $0$ to $3$. The process of evaluating the series involves the following knowledge points:
Exponentiation: Understanding how to raise a number to a power, which is a basic operation in mathematics. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$.
Geometric Series: This is 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. In this case, the common ratio is $2$.
Summation Notation: The use of the summation symbol $\Sigma$ to represent the sum of a sequence of numbers. The series is evaluated by substituting each integer value of $n$ into the expression to the right of the summation symbol and then adding the results.
Simplification: Combining like terms and performing arithmetic operations to reduce an expression to a simpler form or a single numerical value.
By applying these concepts, one can systematically evaluate the series by expanding, simplifying, and then summing the terms.
