Problem 92 - Algebra 2

Question

Find the 100 th term of the arithmetic sequence \(3,10,17,24,31, \ldots\) Explain your steps.

Step-by-Step Solution

Answer

The 100th term of the arithmetic sequence is 696.

Step 1: Identify the first term and common difference

The first term \( a \) is 3. The common difference \( d \) in an arithmetic sequence is found by subtracting the first term from the second term, which gives us \( d = 10 - 3 = 7 \). So, the common difference is 7.

Step 2: Apply the nth term formula

We need to find the 100th term. This means \( n = 100 \). Now we can use the formula for the nth term of an arithmetic sequence: \( a_n = a + (n-1)d \). Substituting our values, we get: \( a_100 = 3 + (100-1)*7 \).

Step 3: Calculate the 100th term

Perform the operations to obtain the 100th term: \( a_100 = 3 + 99*7 = 3 + 693 = 696 \).

Key Concepts

Common Difference

In arithmetic sequences, the common difference is a key element that helps determine the pattern of the sequence. It is the difference between any two successive terms. Understanding the common difference is essential as it tells us how much the sequence increases or decreases by each step.

To find the common difference in any arithmetic sequence:

• Look at the first few terms of the sequence.
• Subtract the first term from the second term.

For example, in the sequence 3, 10, 17, 24, 31, the common difference is calculated as: \[ d = 10 - 3 = 7 \] This means each number is 7 more than the previous number. Knowing the common difference helps in identifying the sequence and solving further problems related to it.

Nth Term Formula

The nth term formula is an invaluable tool for working with arithmetic sequences. It allows us to find any term in the sequence without having to list all the terms up to that point. The formula for the nth term, denoted as \( a_n \), is: \[ a_n = a + (n-1)d \] Here:

• \( a \) is the first term of the sequence.
• \( d \) is the common difference.
• \( n \) is the term number you want to find.

For instance, to find the 100th term in the sequence 3, 10, 17, 24, 31:

• First term \( a = 3 \).
• Common difference \( d = 7 \).
• Plug these into the formula: \( a_{100} = 3 + (100-1) \times 7 \).
• Calculate to find \( a_{100} = 696 \).

This process allows you to skip directly to any term, making it an efficient way to solve sequence problems.

Sequence Problems

Sequence problems often involve identifying patterns and applying formulas to find specific terms. When dealing with arithmetic sequences, you're frequently tasked with questions like finding the nth term or summing series. Understanding how to correctly apply the concepts of common difference and nth term formula is crucial.

Here’s a strategy to solve sequence problems:

• First, identify the sequence as arithmetic by confirming a constant common difference.
• Use the nth term formula to find any term, including distant terms in the sequence that aren’t immediately clear.
• If asked to find the sum, consider using formulas related to arithmetic series.

A common challenge students face is misidentifying the sequence or incorrectly calculating the common difference. Make sure to double-check these early steps before applying further formulae. With practice, these problems become more intuitive, and recognizing patterns becomes second nature.