Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Express the sum in terms of summation notation. (Answers are not unique.) $$2+4+8+16+32+64+128$$

Step-by-Step Solution

Answer

The sum can be expressed as \(\sum_{n=1}^{7} 2^n\).

Step 1: Identifying the Pattern

First, observe the list of numbers: 2, 4, 8, 16, 32, 64, 128. Notice that each number, starting from 2, is doubled from the previous number. This is a geometric sequence where each term is twice the previous one.

Step 2: Determine the Formula for Each Term

Recognize that each number can be represented as a power of 2. Specifically, the numbers are: \(2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7\). Notice that if you start the indexing from 1, the nth term is \(2^n\).

Step 3: Summation Notation

To express the sum using summation notation, we can write it as the sum of \(2^n\) for values of \(n\) from 1 to 7. In summation notation, this is expressed as: \[\sum_{n=1}^{7} 2^n\]

Step 4: Verification of Terms

Double-check to ensure that when \(n=1\), we start at 2 (\(2^1\)), and when \(n=7\), we end at 128 (\(2^7\)). This confirms that our summation notation correctly represents the series.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." In our exercise, the sequence 2, 4, 8, 16, 32, 64, 128 is a perfect example of a geometric sequence. Here, each term is obtained by multiplying the previous term by 2.

This specific kind of geometric sequence, where the common ratio is 2, is often referred to as a doubling sequence. This means each successive term in the sequence is double the previous term:

• Starting term: 2
• Common ratio: 2

Understanding geometric sequences helps in predicting future terms and finding the sum of the sequence using summation notation. It's the stepping stone to comprehending how series representation works as well.

Powers of 2

The phrase "powers of 2" refers to numbers expressed as exponents of 2. These are fundamental in both mathematics and computer science. In our scenario, each term in the series can be represented as a power of 2:

• 2 is written as \(2^1\)
• 4 is written as \(2^2\)
• 8 is written as \(2^3\)
• 16 is written as \(2^4\)
• 32 is written as \(2^5\)
• 64 is written as \(2^6\)
• 128 is written as \(2^7\)

When recognizing these numbers as powers of 2, we see a pattern where the base remains constant, but the exponent increases incrementally. This makes it simple to express the entire series in terms of exponents, facilitating calculations in summation notation.

Series Representation

Series representation is a succinct way to display and compute sums of sequences using formulas and notation. In mathematics, summation notation, often referred to as sigma notation, is a powerful tool to simplify the representation of partial or infinite sums.

In our exercise, representing the series 2 + 4 + 8 + 16 + 32 + 64 + 128 with summation notation involves utilizing the expression of each term as a power of 2, specifically \(2^n\) where \(n\) starts at 1 and ends at 7. Here’s how it's expressed:\[ \sum_{n=1}^{7} 2^n\]This confirms that you're summing from the first term, \(2^1\) or 2, up to \(2^7\) or 128.

By representing our series in this way, we achieve a clearer, more concise form that allows mathematicians to analyze and work with long sequences more efficiently.