Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{15}=8, a_{17}=2$$

Step-by-Step Solution

Answer

\(a_8 = 29\) and \(a_n = 53 - 3n\).

Step 1: Find the Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant. Let's use the given terms, \(a_{15} = 8\) and \(a_{17} = 2\), to find this common difference \(d\). \[\begin{align*} a_{17} &= a_{15} + 2d, \ 2 &= 8 + 2d. \end{align*}\] Solve for \(d\): \[2 - 8 = 2d \implies -6 = 2d \implies d = -3.\] So, the common difference is \(d = -3\).

Step 2: Write the General Formula for the Sequence

Since we have the common difference, we can write the general term for the sequence. The \(n\)-th term of an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\). We can use the known terms to find \(a_1\).

Step 3: Find the First Term \(a_1\)

Using \(a_{15} = 8\), we have: \[\begin{align*} a_{15} &= a_1 + 14d, \ 8 &= a_1 + 14(-3). \end{align*}\] Solving for \(a_1\): \[\begin{align*} 8 &= a_1 - 42, \ a_1 &= 8 + 42, \ a_1 &= 50. \end{align*}\] The first term is \(a_1 = 50\).

Step 4: Find \(a_8\)

Now that we have \(a_1\) and the common difference \(d\), let's find \(a_8\). The formula for any term is \(a_n = a_1 + (n-1)d\): \[\begin{align*} a_8 &= 50 + 7(-3), \ &= 50 - 21, \ &= 29. \end{align*}\] Therefore, \(a_8 = 29\).

Step 5: Express \(a_n\)

Using the general term formula, we express \(a_n\) with \(a_1 = 50\) and \(d = -3\): \[a_n = 50 + (n-1)(-3).\] This simplifies to: \[a_n = 50 - 3(n-1)\] or \[a_n = 53 - 3n.\]

Key Concepts

Common Difference

In an arithmetic sequence, the magic ingredient is the "common difference." This is the constant difference between consecutive terms in the sequence. Each term in the sequence is obtained by adding this constant value to the previous term.
For the sequence in our example, we were given two terms: \(a_{15} = 8\) and \(a_{17} = 2\). Using these terms, we can calculate the common difference \(d\).

• The formula to calculate the common difference is \(d = \frac{a_{17} - a_{15}}{17 - 15}\).
• So, \(d = \frac{2 - 8}{2} = -3\).

This shows us that in this sequence, each term is 3 less than the one before it. This consistent difference helps us in forming other terms and working with the sequence.

General Formula

Once we find the common difference, we can write the general formula for the sequence. This formula is a blueprint that helps us calculate any term in the sequence without needing to list every single term.
The general formula for an arithmetic sequence, where \(a_1\) is the first term and \(d\) is the common difference, is given by:
\[a_n = a_1 + (n-1)d\]
This formula tells us that to find the \(n\)-th term \(a_n\), we start from the first term \(a_1\) and add the common difference \(d\) multiplied by \((n-1)\).
It becomes a powerful tool once we know the first term and common difference since it allows us to calculate any term directly.

First Term

The first term in an arithmetic sequence, denoted as \(a_1\), is the starting point from which all other terms in the sequence are built. In our problem, we need to find \(a_1\) to use it in the general formula.
We use the provided \(a_{15}\) and the common difference we calculated earlier to find \(a_1\):

• Using the formula, we have \(a_{15} = a_1 + 14d\).
• Substituting \(a_{15} = 8\) and \(d = -3\), we solve: \(8 = a_1 + 14(-3)\).
• This simplifies to \(8 = a_1 - 42\), thus \(a_1 = 50\).

By finding \(a_1 = 50\), we establish the foundation for our sequence. This step is essential as it links the known terms with the unknown ones.

Term Calculation

Now that we know both the first term \(a_1\) and the common difference \(d\), we can calculate any term in the arithmetic sequence using the general formula.
For instance, to find the eighth term \(a_8\), we use:

• The formula \(a_n = a_1 + (n-1)d\).
• Plug in \(n = 8\) to get \(a_8 = 50 + 7(-3)\).
• This simplifies to \(a_8 = 50 - 21 = 29\).

Similarly, we can express the sequence's \(n\)-th term. By applying the values of \(a_1\) and \(d\) into the general formula, we get \(a_n = 53 - 3n\).
This expression is incredibly useful for identifying any term you want in the sequence with just a plug-in of \(n\).