Problem 53 - Precalculus with Limits

Question

In Exercises 51 - 58, find the sum of the finite arithmetic sequence. \( -1 + \left(-3\right) + \left(-5\right) + \left(-7\right) + \left(-9\right) \)

Step-by-Step Solution

Answer

The sum of the given arithmetic sequence is -25.

Step 1: Identify the first, last term and the number of terms

The first term, \(a_1\), of the sequence is -1. The last term, \(a_n\), is -9 and the sequence consists of 5 terms in total.

Step 2: Add the terms manually

The sum of the arithmetic sequence is found by adding the terms: \( S_n = -1 + -3 + -5 + -7 + - 9 = -25 \)

Step 3: Calculate the sum via the formula

Let's also verify the previous result using the formula for the sum of an arithmetic sequence: \( S_n = \frac{n}{2} \cdot (a_1 + a_n) = \frac{5}{2} \cdot (-1 + -9) = -25 \)

Key Concepts

Sum of a Sequence

The sum of a sequence can feel like adding multiple numbers together, which it essentially is! The sum of an arithmetic sequence, in particular, is very straightforward once you understand the basic concept. To find this sum, you add up all the terms in the sequence. However, you can make it even easier by using a formula designed specifically for arithmetic sequences. This is where the wonder of mathematics saves you from manual calculations each time.

In the given example, we are dealing with an arithmetic sequence, so the formula for the sum of such a sequence is used to make the process quicker and more accurate.

• Manually, you add each term: -1 + (-3) + (-5) + (-7) + (-9) = -25.
• Using the formula, we get the same result: \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \).

This formula comes in handy, especially with larger sequences or when you prefer avoiding the hassle of adding each number manually.

Finite Sequence

A finite sequence is simply a sequence that has a limited number of terms. Unlike infinite sequences that continue indefinitely, finite sequences end after a certain point. In our example, the sequence has five terms: -1, -3, -5, -7, and -9.

Knowing a sequence is finite helps you determine certain properties such as the number of terms (often represented as \( n \)) and allows you to apply formulas effectively. In this case, \( n = 5 \), indicating there are 5 terms to be concerned with. This knowledge greatly aids in using the sum formula for arithmetic sequences.

• Finite sequences are generally easier to handle since you know exactly how many elements you're dealing with.
• With a finite sequence in arithmetic progression, calculating the total sum is direct once you identify the sequence’s type and length.

Understanding the limits of sequence terms ensures clarity in mathematical calculations and applications.

Arithmetic Sequence Formula

The arithmetic sequence formula is a lifesaver when you're dealing with arithmetic progressions. In these sequences, each term increases or decreases at a constant value known as the common difference. For our example of -1, -3, -5, -7, -9, the common difference is -2 (since each term is obtained by subtracting 2 from the previous one).

There are crucial components to remember:

• The first term \( a_1 \) (here, -1)
• The last term \( a_n \) (here, -9)
• The number of terms \( n \) (here, 5)

To find the sum of this arithmetic sequence, you apply the formula \( S_n = \frac{n}{2} \cdot (a_1 + a_n) \). Meaning, you multiply the average of the first and last terms by the number of terms divided by 2. Using this formula simplifies the process exponentially, especially for longer sequences.This practice ensures accurate results quickly, affirming its place in your mathematical toolkit.