Problem 34 - Algebra for College Students

Question

If the 5 th term of an arithmetic sequence is \(-5\) and the 15 th term is \(-25\), find the 50 th term. \(-95\)

Step-by-Step Solution

Answer

The 50th term is -95.

Step 1: Understanding the Problem

The problem is asking for the 50th term of an arithmetic sequence where we already know the 5th and 15th terms. An arithmetic sequence has a common difference between consecutive terms. We need to find the common difference and then use it to find the 50th term.

Step 2: Identifying Known Values

In the arithmetic sequence, the 5th term is given as \(a_5 = -5\) and the 15th term is \(a_{15} = -25\). We can use these terms to find the common difference.

Step 3: Finding the Common Difference

The general formula for the nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term and \(d\) is the common difference. Using the given terms, we can set up the following equations:1. \(a_5 = a_1 + 4d = -5\)2. \(a_{15} = a_1 + 14d = -25\)Subtract the first equation from the second to find \(d\): \[(a_1 + 14d) - (a_1 + 4d) = -25 - (-5)\]\[10d = -20\]\[d = -2\]

Step 4: Finding the First Term

Now that we have the common difference \(d = -2\), we can find the first term \(a_1\) using one of the known terms. Using \(a_5 = a_1 + 4(-2) = -5\):\(a_1 - 8 = -5\)\(a_1 = 3\)

Step 5: Calculating the 50th Term

Now we know \(a_1 = 3\) and \(d = -2\), we can find the 50th term using the formula for the nth term:\(a_{50} = a_1 + (50-1) imes d\)\(a_{50} = 3 + 49 imes (-2)\)\(a_{50} = 3 - 98\)\(a_{50} = -95\)

Step 6: Conclusion

We have calculated the 50th term of the arithmetic sequence, which is \(-95\).

Key Concepts

Common Difference

In an arithmetic sequence, the term "common difference" is key. It's the consistent amount we add (or subtract) to get from one term to the next. To find this number, we use any two consecutive terms in the sequence.
For instance, in the provided problem, we have a situation where we're given the 5th term and the 15th term. By using these two known terms, we can identify the common difference:

• Subtract the known earlier term from the known later term
• Divide by the number of jumps between the two terms

Here’s how it worked in our example:
The 15th term is (-25) and the 5th term is (-5). Since there are 10 terms between them, we simply subtract (-5) from (-25) which gives us (-20), then divide this by 10. Thus, the common difference (d) is (-2).
Understanding this helps in predicting any term of the sequence using the common difference.

Nth Term Formula

The nth term formula is essential when dealing with arithmetic sequences. It allows us to find any specific term without having to list all the terms.
It follows this general format:
\[a_n = a_1 + (n-1) \times d\]Where:

• \(a_n\) is the term we seek.
• \(a_1\) is the first term.
• \(d\) is the common difference.
• \(n\) is the term's position in the sequence.

In our example, once we found \(a_1 = 3\) and \(d = -2\), we used this formula to calculate the 50th term, \(a_{50}\). By plugging these values into our formula, we determined that \(a_{50} = -95\). This demonstrates how powerful and efficient the nth term formula can be in arithmetic sequences.

Sequence Terms Identification

Identifying terms within an arithmetic sequence is all about using the known information to uncover the unknown terms. It's like building a puzzle where certain pieces (the known terms) guide us to complete the image (the rest of the sequence).
In arithmetic sequences, we know that each term relates directly to the next via the common difference. Once we have the common difference, identifying any term from any part of the sequence becomes manageable.
Steps for sequence recognition:

• Establish the common difference using known terms.
• Apply the nth term formula to find the desired term.

When you have terms like the 5th and the 15th, as in our example, these help us work backward or forward to identify the first term or any other term needed. Once you have the first term (\(a_1\)), and the common difference (\(d\)), recognizing every term is straightforward by plugging into the nth term formula. With practice, this process becomes quite intuitive!