Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{10} ; \quad a_{2}=1, \quad a_{18}=49 $$

Step-by-Step Solution

Answer

The 10th term, \(a_{10}\), is 25.

Step 1: Understand the Problem

We are given two terms of an arithmetic sequence: \(a_2 = 1\) and \(a_{18} = 49\). We need to find the 10th term, \(a_{10}\), of this sequence.

Step 2: Use the General Formula for Arithmetic Sequence

The formula to find the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. We will use this information to find \(d\) and subsequently find \(a_{10}\).

Step 3: Set Up Equations for Known Terms

Using the general formula, we have:1. For \(a_2\): \(a_2 = a_1 + d = 1\)2. For \(a_{18}\): \(a_{18} = a_1 + 17d = 49\) We will solve these two equations to find \(a_1\) and \(d\).

Step 4: Solve for Common Difference \(d\)

Subtract equation 1 from equation 2:\[(a_1 + 17d) - (a_1 + d) = 49 - 1\]This simplifies to:\[16d = 48\]Solving for \(d\), we get:\[d = \frac{48}{16} = 3\]

Step 5: Solve for First Term \(a_1\)

Substitute \(d = 3\) back into the equation from Step 3 for \(a_2\):\[a_1 + 3 = 1\]Solving for \(a_1\), we get:\[a_1 = 1 - 3 = -2\]

Step 6: Find the 10th Term \(a_{10}\)

Using \(a_1 = -2\) and \(d = 3\), we use the formula for the 10th term:\[a_{10} = a_1 + 9d = -2 + 9 \times 3\]Calculating this, we find:\[a_{10} = -2 + 27 = 25\]

Key Concepts

nth Term Formula

The formula to find the nth term of an arithmetic sequence is crucial for solving these types of problems. It's defined as \(a_n = a_1 + (n-1)d\).
This formula allows you to determine any term in a sequence if the first term \(a_1\) and the common difference \(d\) are known.
Why is this important? It gives a systematic way to calculate terms without listing the entire sequence. For instance, if you know \(a_1\) and \(d\), calculating any term, say \(a_{10}\), becomes straightforward by plugging \(n = 10\) into the formula.
Remember that in arithmetic sequences, the difference between consecutive terms is always constant, reinforcing the linear nature captured in the formula.

Common Difference

The common difference \(d\) is a key feature of an arithmetic sequence. It is the consistent amount added (or subtracted) as you move from one term to the next.
You can compute \(d\) by subtracting one term in the sequence from the next. In our exercise, we see this in action:
For given terms \(a_2 = 1\) and \(a_{18} = 49\), we set up the following equations based on the general formula:

• \(a_2 = a_1 + d = 1\)
• \(a_{18} = a_1 + 17d = 49\)

By subtracting these equations, you isolate \(d\).
It’s important because without \(d\), the sequence would lack structure, and the nth term formula wouldn't apply.

Solving Linear Equations

Solving linear equations is an essential skill for finding the unknowns in arithmetic sequences.
In the step-by-step solution, after establishing equations from the terms provided, we proceed with solving for the common difference \(d\) and subsequently the first term \(a_1\).
Linear equations come into play to eliminate unknowns and derive solutions concretely. Using the example:

• The difference equation \(16d = 48\) straightforwardly yields \(d = 3\).
• Substituting \(d = 3\) back into \(a_2 = a_1 + d = 1\) allows us to find \(a_1 = -2\).

Always keep your steps methodical: set up your equations clearly, isolate the variable you're solving for, and ensure each step follows logically from the previous.
Being proficient in solving these equations ensures you can tackle a variety of sequence-related problems with confidence.