Problem 33 - College Algebra

Question

Find the indicated sum. $$ \sum_{k=1}^{7} 2^{k} $$

Step-by-Step Solution

Answer

The sum is 254.

Step 1: Understand the Expression

The given expression \( \sum_{k=1}^{7} 2^{k} \) represents a summation where each term is of the form \( 2^k \) and the variable \( k \) runs from 1 to 7.

Step 2: Set Up the Sum

Write out each term in the summation: \( 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, \text{ and } 2^7.\)

Step 3: Calculate Each Term

Compute the value of each power of 2: \( 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, 2^5 = 32, 2^6 = 64, 2^7 = 128. \)

Step 4: Sum the Terms

Add the calculated values together: \( 2 + 4 + 8 + 16 + 32 + 64 + 128.\) Simplifying gives \( 254.\)

Step 5: Verify the Solution

Re-check each calculation of the powers of 2, and re-add the terms to ensure the arithmetic is correct, confirming the sum is \( 254.\)

Key Concepts

Understanding Powers of 2

The concept of powers of 2 is fundamental in understanding exponential growth in mathematics. When we talk about powers of 2, we are referring to numbers in the form of \(2^k\), where \(k\) is any integer. Here's how it works:

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)

This pattern continues exponentially because each power is the previous one multiplied by 2. Incomputing and digital systems, powers of 2 are particularly significant. This is because binary systems, the basis of computer operations, use powers of 2 extensively.
For instance, a byte comprises 8 bits, representing up to \(2^8\) (256) different values. Understanding how quickly powers of 2 grow helps in solving problems related to exponential sequences and growth patterns.

Exploring Arithmetic Series

An arithmetic series is a sequence of numbers with a constant difference between consecutive terms. However, in this exercise, the terms represent powers of 2, which do not form an arithmetic series. Instead, what we compute is the sum of consecutive terms of an exponential function. It's important to differentiate between the two concepts:

In arithmetic series:

• Terms are added by a constant value.
• Examples include sequences like 2, 4, 6, 8,... adding 2 each time.

In contrast, our given sequence is 2, 4, 8, 16,... which doubles each time addressing exponential growth. Recognizing parallels between the two concepts allows us to apply sum formulas efficiently in different mathematical contexts. These series help in problem-solving branches of finance, engineering, and more.

Sequences in Mathematics

Sequences are ordered lists of numbers, and understanding them is crucial in mathematics. Each number in a sequence is called a "term." Sequences can either be finite, like our example of summing from \(k=1\) to \(k=7\), or infinite, where numbers continue endlessly. Within sequences, recognizing patterns or rules governing the terms is key to solving mathematical problems:

• Arithmetic sequences, where terms increase by a constant amount.
• Geometric sequences, where each term multiplies by a constant (e.g., powers of 2).

Our original problem featured a sequence where each term was derived as a power of 2, thus a geometric sequence. Knowing the structure of sequences helps in identifying summing strategies and in establishing how terms interrelate. Sequences are useful in predicting future values and assessing growth over intervals in varied scientific and financial fields.