Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{1}=5, d=-\frac{3}{4}$$

Step-by-Step Solution

Answer

The first 5 terms of the given arithmetic sequence are 5, 4.25, 3.5, 2.75, 2.

Step 1: Identify the first term and the common difference.

In this problem, the first term 'a1' and the common difference 'd' are given. The first term \(a_{1}\) is 5, and the common difference 'd' is -3/4.

Step 2: Find the second term of the sequence.

The formula for the nth term of an arithmetic sequence is \(a_{n} = a_{1} + (n-1)d\). To find the second term, n should be 2. So \(a_{2} = a_{1} + (2-1)d = 5 + 1*(-3/4) = 5 - 3/4 = 4.25\).

Step 3: Find the third term of the sequence.

Using the same formula, when n is equal to 3, \(a_{3} = a_{1} + (3-1)d = 5 + 2*(-3/4) = 5 - 1.5 = 3.5\).

Step 4: Find the fourth term of the sequence.

Again using the formula, when n equals 4, \(a_{4} = a_{1} + (4-1)d = 5 + 3*(-3/4) = 5 - 2.25 = 2.75\).

Step 5: Find the fifth term of the sequence.

Using the formula for nth term of the sequence, when n equals 5, \(a_{5} = a_{1} + (5-1)d = 5 + 4*(-3/4) = 5 - 3 = 2\).

Key Concepts

Understanding the First Term

In arithmetic sequences, the first term is the starting number of the series. It acts as a baseline from which all subsequent terms are calculated. Given our sequence, the first term is denoted as \( a_1 \) and is valued at 5.
The first term establishes the initial value and is crucial for determining future terms using the arithmetic sequence formula.
This foundation allows you to compute any term in the sequence, provided you know the common difference.

Exploring the Common Difference

The common difference in an arithmetic sequence is the consistent amount each term increases or decreases from one term to the next. In our sequence example, the common difference is \( d = -\frac{3}{4} \).
This term is crucial since it dictates how the sequence changes:

• If \( d \) is positive, each term is larger than the last, resulting in a rising sequence.
• If \( d \) is negative, like in this case, the sequence decreases, producing steadily smaller terms.

Understanding and identifying the common difference allows you to predict the pattern of the sequence effectively.

Decoding the nth Term Formula

The nth term formula in an arithmetic sequence helps compute the value of any term given its position, known as \( n \). The formula is expressed as:
\[ a_n = a_1 + (n-1) \cdot d \]Here’s a simple breakdown:

• \( a_n \): Represents the term you are trying to find.
• \( a_1 \): Is the first term of the sequence.
• \( n \): Denotes which term number you are calculating.
• \( d \): Indicates the common difference between the terms.

This formula is versatile and helps calculate any term in the sequence efficiently, whether it's the second or the fiftieth term. Just plug in the values you have to find the desired term.