Problem 16 - Algebra for College Students

Question

\text { Find the indicated sum } \sum_{i=1}^{10}(-2)^{i-1}

Step-by-Step Solution

Answer

The sum is -341.

Step 1: Understand the Sum Notation

The notation \( \sum_{i=1}^{10} (-2)^{i-1} \) indicates that we need to calculate the sum of a sequence of terms generated by the expression \((-2)^{i-1}\), starting from \(i = 1\) to \(i = 10\).

Step 2: Generate the Sequence

Create the sequence by substituting values of \(i\) from \(1\) to \(10\) into the expression \((-2)^{i-1}\):- For \(i = 1\), \((-2)^{1-1} = 1\)- For \(i = 2\), \((-2)^{2-1} = -2\)- For \(i = 3\), \((-2)^{3-1} = 4\)- For \(i = 4\), \((-2)^{4-1} = -8\)- For \(i = 5\), \((-2)^{5-1} = 16\)- For \(i = 6\), \((-2)^{6-1} = -32\)- For \(i = 7\), \((-2)^{7-1} = 64\)- For \(i = 8\), \((-2)^{8-1} = -128\)- For \(i = 9\), \((-2)^{9-1} = 256\)- For \(i = 10\), \((-2)^{10-1} = -512\)

Step 3: Calculate the Sum

Add up all the terms from the sequence obtained in Step 2:\[ 1 + (-2) + 4 + (-8) + 16 + (-32) + 64 + (-128) + 256 + (-512) \]Simplify by performing the arithmetic:\[ = 1 - 2 + 4 - 8 + 16 - 32 + 64 - 128 + 256 - 512 \]\[ = -341 \]

Key Concepts

Sequence

A sequence is a list of numbers in a specific order. Each number in the sequence is called a term. In this exercise, the sequence is generated by the expression \((-2)^{i-1}\). The sequence starts when \(i=1\) and continues through \(i=10\).
In this context, our sequence is:

• For \(i=1\), the term is \((-2)^{0} = 1\)
• For \(i=2\), the term is \((-2)^{1} = -2\)
• For \(i=3\), the term is \((-2)^{2} = 4\)
• And so forth until \(i=10\)

This sequence alternates between positive and negative numbers because the base \(-2\) is raised to consecutive integers, flipping the sign of each successive term.
Sequences can describe patterns or structures in mathematical problems. In this case, it helps us identify the individual elements we need to sum to find the solution.

Arithmetic Series

An arithmetic series is defined as the sum of the terms of an arithmetic sequence. However, it's important to note that our current sequence is not an arithmetic one. In arithmetic sequences, the difference between consecutive terms stays constant.
Instead, this exercise involves a geometric nature due to the exponentiation of the base \(-2\), making use of exponent rules rather than constant differences.
To understand such series, make a note of how the terms grow or diminish. Here, it grows dramatically because of the exponentiation, resulting in widely varying terms. Thus, while this isn't an arithmetic series, understanding the concept helps differentiate types of sequences and series.

Exponentiation

Exponentiation involves raising a number (the base) to the power of an exponent. It is a powerful mathematical operation that greatly affects the size of numbers.
In the given problem, the formula \((-2)^{i-1}\) demonstrates exponentiation. The base is \(-2\) and the exponent \(i-1\) changes as \(i\) increments from \(1\) to \(10\). This process creates a unique sequence of terms which we eventually sum.
Key points about exponentiation include:

• A positive base with an even exponent results in a positive number.
• A positive base with an odd exponent results in a negative number.
• For a negative base, these rules flip accordingly.

Understanding exponentiation helps solve the problem by generating each term in the sequence based on the value of \(i\). In arithmetic, it allows us to calculate very large or very small numbers effectively by employing powers.