Problem 35 - 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. $$a_{8}=26, a_{12}=42$$

Step-by-Step Solution

Answer

The first five terms of the sequence are -2, 2, 6, 10 and 14.

Step 1: Determining the Common Difference

In an arithmetic sequence, the common difference \(d\) can be found by subtracting any term (except the first) from the one following it.Here we are given \(a_{8}=26\) and \(a_{12}=42\). The common difference can be found using the formula: \(d = \frac{a_{n} - a_{m}}{n - m}\)So, \(d = \frac{a_{12} - a_{8}}{12 - 8} = \frac{42 - 26}{4} = 4\). Hence, the common difference \(d\) is 4.

Step 2: Finding the First Term (a1)

The nth term of an arithmetic sequence can be calculated by the formula \(a_{n} = a_{1} + (n-1) * d\). In this case, you know \(a_{8}\) = 26, with \(n = 8\) and \(d = 4\).Solving for \(a_{1}\), we get: \(a_{1} = a_{n} - (n-1) * d = 26 - (8-1) * 4 = 26 - 28 = -2\). Thus, the first term \(a_{1}\) of the sequence is -2.

Step 3: Calculating the first five terms

Now, using the first term and the common difference, the first five terms can be calculated as:Term 1 (a1): -2Term 2 (a2): \(a_{1} + d = -2 + 4 = 2\)Term 3 (a3): \(a_{2} + d = 2 + 4 = 6\)Term 4 (a4): \(a_{3} + d = 6 + 4 = 10\)Term 5 (a5): \(a_{4} + d = 10 + 4 = 14\)

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a vital component that determines the progression from one term to the next. To find the common difference, you simply subtract a term from the term following it. This subtraction should be consistent across the sequence.
For this exercise, we know that the terms provided are \( a_8 = 26 \) and \( a_{12} = 42 \). By applying the formula for the common difference \( d \) which is \( d = \frac{a_{n} - a_{m}}{n - m} \), we calculate:

• \( d = \frac{42 - 26}{12 - 8} \)
• \( d = \frac{16}{4} = 4 \)

Thus, the common difference \( d \) is 4. This means each term in the sequence increases by 4 from the previous one.

First Term of Sequence

The first term of an arithmetic sequence is the starting point that helps in building the rest of the sequence using the common difference. Knowing the first term is crucial as it sets the entire sequence into motion.
To find the first term \( a_1 \), we use the nth-term formula for an arithmetic sequence:

• \( a_{n} = a_{1} + (n-1) \cdot d \)
• Given: \( a_8 = 26 \), \( n = 8 \), \( d = 4 \)

The calculation for \( a_1 \) is:

• \( a_1 = 26 - (8-1) \cdot 4 \)
• \( a_1 = 26 - 28 = -2 \)

Therefore, the first term \( a_1 \) of the sequence is \( -2 \). With this, and the knowledge of the common difference, we can determine all subsequent terms.

Graphing Utility Verification

Graphing utilities are helpful tools for verifying the accuracy of your arithmetic sequence calculations. By plotting the sequence on a graph, you can visually confirm the correctness of the calculated terms.
To verify our results using a graphing utility:

• Input the first term \( a_1 = -2 \) and the common difference \( d = 4 \) into the utility.
• Configure the utility to display the first term and calculate subsequent terms by adding the common difference repeatedly.
• Check the table feature to see if the sequence aligns with our calculated terms: \( -2, 2, 6, 10, 14 \).

If the terms match, our calculations are correct. Visual verification via a graph adds an extra layer of confidence to your index calculations.