Problem 47 - Algebra and Trigonometry

Question

\(41-48\) Find the sum. $$ \sum_{k=1}^{5} 2^{k-1} $$

Step-by-Step Solution

Answer

The sum is 31.

Step 1: Understand the Sum Notation

The given expression \( \sum_{k=1}^{5} 2^{k-1} \) is a summation notation. It's asking us to calculate the sum of the series from \( k=1 \) to \( k=5 \), where each term is given by \( 2^{k-1} \).

Step 2: Calculate Each Term

Let's calculate the individual terms for each \( k \): - When \( k=1 \), \( 2^{1-1} = 2^0 = 1 \).- When \( k=2 \), \( 2^{2-1} = 2^1 = 2 \).- When \( k=3 \), \( 2^{3-1} = 2^2 = 4 \).- When \( k=4 \), \( 2^{4-1} = 2^3 = 8 \).- When \( k=5 \), \( 2^{5-1} = 2^4 = 16 \).

Step 3: Sum the Calculated Terms

Now we add all the terms we calculated: \( 1 + 2 + 4 + 8 + 16 = 31 \).

Step 4: Verify the Result

We can quickly check the calculation by reviewing each term and the addition:- Start with \( 1 \).- Add \( 2 \) to get \( 3 \).- Add \( 4 \) to get \( 7 \).- Add \( 8 \) to get \( 15 \).- Add \( 16 \) to finally get \( 31 \).The calculations confirm that the sum is correct.

Key Concepts

Series

A series is essentially the sum of a sequence of numbers. When we use summation notation (the symbol \( \Sigma \)), it allows us to efficiently represent and calculate the sum of large sequences. In the context of our problem, the series is represented as \( \sum_{k=1}^{5} 2^{k-1} \). Here, \( k \) varies from 1 to 5, and for each value of \( k \), we calculate \( 2^{k-1} \).

• A series can be finite, as in this exercise, where \( k \) runs from 1 to a specific end value (5).
• Series with a clear pattern in the terms, like geometric or arithmetic series, allow for quicker calculations using formulas.
• Each term of this series builds upon the previous one, following the rule given.

The value of the series can be found by computing each term of the sequence, then summing them up, which is exactly what we demonstrated in the solution.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. The notation \( a^b \) signifies that the number \( a \) is multiplied by itself \( b \) times. In the given exercise, exponents are used to calculate the terms of the series: \( 2^{k-1} \).

• The exponentiation operation \( 2^{k-1} \) is crucial because it gives the power of 2 for each value of \( k \).
• Starting from \( 2^0 = 1 \), the series progresses to \( 2^1 = 2 \), then to \( 2^2 = 4 \), and so on.
• This demonstrates the base 2 exponential growth pattern, where each subsequent number in the sequence is twice the previous one.

Understanding how exponents work helps unravel many mathematical patterns and problems, especially those involving growth, such as compounded interest or population studies.

Algebra

Algebra involves the manipulation of symbols and formulas to solve equations and understand functions. In our series problem, algebra helps us decode the summation notation and calculate each term.

• We use algebraic manipulation to understand the terms of the series and set up calculations logically.
• The recursive pattern, \( k=1 \) to \( k=5 \), shows how algebra can model repetitive processes with simplicity.
• Applying basic algebra to sum these terms ensures accuracy and understanding in evaluating expressions.

Algebra is foundational in mathematics because it allows for the generalization of arithmetic operations and paves the way for solving complex equations.