Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated sum. \(\sum_{k=1}^{7} 2^{k}\)

Step-by-Step Solution

Answer

The sum is 254.

Step 1: Understand the Problem

The exercise asks us to find the sum of the powers of 2 from 1 to 7, which is represented as \( \sum_{k=1}^{7} 2^{k} \). This means we must add up all the values of \( 2^k \) for \( k = 1, 2, 3, \ldots, 7 \).

Step 2: Calculate Each Term

Start by calculating the powers of 2 for each integer \( k \) from 1 to 7. Here are the calculations: - \( 2^1 = 2 \)- \( 2^2 = 4 \)- \( 2^3 = 8 \)- \( 2^4 = 16 \)- \( 2^5 = 32 \)- \( 2^6 = 64 \)- \( 2^7 = 128 \)

Step 3: Sum the Terms

Sum up all the terms calculated in the previous step:\[ 2 + 4 + 8 + 16 + 32 + 64 + 128 \].Performing the addition gives \[ 2 + 4 + 8 + 16 + 32 + 64 + 128 = 254 \].

Step 4: Verify the Solution

To verify the solution, ensure that each calculated term is correct and that no numbers in the addition were missed. Re-calculating:\(2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64,\) and \(2^7 = 128\), so the completed sum \( 254 \) is confirmed.

Key Concepts

Understanding Powers of Two

Powers of two are a series of numbers where each value represents two raised to the power of a non-negative integer. This series is fundamental in mathematics, especially in binary systems used in computing and digital electronics.

• The notation for a power of two is written as \(2^n\), where \(n\) is the exponent indicating how many times 2 is multiplied by itself.
• For example, \(2^3\) means \(2 \times 2 \times 2 = 8\).
• Exponentiating 2 allows quick calculations, especially when dealing with binary data which fundamentally relies on sequences of zeros and ones (bits).

Understanding these powers is crucial. It allows us to simplify expressions and solve various mathematical problems efficiently, like the one given in the exercise above.

Exploring Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This can be a handy tool in problem-solving as it helps in finding sums quickly and logically.

• In the context of the powers of two, each term in the series increases in a non-uniform way because of the exponential nature of powers.
• Although the sum \( \sum_{k=1}^{7} 2^{k} \) is not an arithmetic series by itself, understanding series helps grasp problems with patterns and sequences.
• The general formula for the sum \( S \) of the first \( n \) terms of an arithmetic series is \( S = \frac{n}{2} (a + l) \), where \( a \) is the first term, and \( l \) is the last term.

For this specific problem, knowing this pattern would not directly apply but is essential for broader comprehension in mathematical problem solving.

Problem Solving in Mathematics

Mathematical problem solving involves several steps including understanding, planning, and executing the solution, and checking the answer. These steps help ensure the accuracy and completeness of the solution.

• Understanding the Problem: It involves breaking down what is being asked, such as identifying that the given sum \( \sum_{k=1}^{7} 2^{k} \) means adding up powers from 1 to 7.
• Planning and Execution: This can involve calculating each power of two, as shown in the solution, and systematically adding these numbers.
• Verification: Re-checking calculated powers and summation to make sure no error has been made, an essential step to catch potential mistakes.

Building a habit around structured problem-solving techniques is invaluable, not only for algebraic problems but for all mathematical situations encountered.