Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

\(\sum_{k=1}^{5}(-3)^{k-1} \quad\)

Step-by-Step Solution

Answer

The sum is 61.

Step 1: Understand the Summation

We need to find the sum \( \sum_{k=1}^{5} (-3)^{k-1} \). This expression tells us that we are summing the sequence of terms that are in the form \((-3)^{k-1}\) for \(k = 1\) to \(k = 5\).

Step 2: Calculate Each Term

Compute each term using the expression \((-3)^{k-1}\) for values of \(k\) from 1 to 5:- When \(k = 1\), the term is \((-3)^{1-1} = (-3)^0 = 1\).- When \(k = 2\), the term is \((-3)^{2-1} = (-3)^1 = -3\).- When \(k = 3\), the term is \((-3)^{3-1} = (-3)^2 = 9\).- When \(k = 4\), the term is \((-3)^{4-1} = (-3)^3 = -27\).- When \(k = 5\), the term is \((-3)^{5-1} = (-3)^4 = 81\).

Step 3: Sum the Terms

Add together all the terms we calculated:\[1 + (-3) + 9 + (-27) + 81 = 61\].

Step 4: State the Result

The result of the summation \( \sum_{k=1}^{5} (-3)^{k-1} \) is 61.

Key Concepts

Geometric Series

A geometric series is a fascinating concept in mathematics where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula to find the sum of a geometric series, especially when the series is finite, is given as \[ S_n = a rac{1 - r^n}{1 - r} \]where

• \( S_n \) is the sum of the series,
• \( a \) is the first term,
• \( r \) is the common ratio, and
• \( n \) is the number of terms.

In our exercise, while not given the series in the initial typical form, each term being \((-3)^{k-1}\) suggests a structure where each term multiplies by -3, although the powers offset the straightforward formula usage. Understanding geometric series makes solving such expressions easier in complex math problems.

Sequence

A sequence is simply an ordered list of numbers. Each number in the list is called a term. Sequences can follow different rules or formulas to determine what each term will be. For instance, in an arithmetic sequence, each term is generated by adding a constant difference to the previous term. In contrast, our exercise used a geometric sequence, where each term is multiplied by a fixed ratio compared to the previous term.

When examining a sequence, take note of the rule governing it, which can often be expressed as a formula. In the exercise, the sequence rule \((-3)^{k-1}\) creates a distinct pattern due to the alternating signs that result from raising a negative number to successive integer powers. This patterning aids in predicting the future terms and their values effectively and is integral for identifying the series' characteristics.

Powers of Negative Numbers

Understanding the behavior of negative numbers when raised to various powers is vital. A negative number raised to an even power will result in a positive number, because multiplying a negative number an even number of times cancels out its negative sign. For instance,

• \((-3)^2 = 9\)
• \((-3)^4 = 81\)

Conversely, raising a negative number to an odd power will keep it negative. For instance,

• \((-3)^1 = -3\)
• \((-3)^3 = -27\)

In our exercise, the pattern of this alternating sign is evident as each power fluctuates between keeping or canceling the negative. This understanding is useful when predicting and computing the outcomes in equations containing negative bases.

Arithmetic Operations

Arithmetic operations are the basic mathematical processes of addition, subtraction, multiplication, and division. They are fundamental and form the building blocks for more complex calculations. In the context of our exercise, arithmetic operations were critical to finding the solution.

Once the individual terms of the sequence were calculated, they needed to be summed together: \[1 + (-3) + 9 + (-27) + 81 = 61\].Here, each addition operation considers not only the numerical value but also the sign, carefully using subtraction where necessary. Utilizing arithmetic operations with precision confirms the final value of the sequence's summation. Mastering these operations and understanding how they fit into larger calculations can significantly aid problem-solving and mathematical intuition.