Problem 22 - Algebra and Trigonometry

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{70}\) when \(a_{1}=-32, d=4\)

Step-by-Step Solution

Answer

The 70th term of the arithmetic sequence is 244.

Step 1: Identify the Given Values

The first term (\(a_{1}\)) is -32 and the common difference (\(d\)) is 4. We are asked to find the term \(a_{70}\), thus \(n=70\).

Step 2: Apply the Formula to Find the Desired Term

We can find the \(70^{th}\) term by using the formula \(a_{n} = a_{1} + (n-1)*d\). Substituting the given values into the formula, we get \(a_{70} = -32 + (70-1)*4\).

Step 3: Calculate the Result

Start calculating the equation set up in step 2, first calculate the value inside the bracket \((70-1) = 69\), multiple this by the common difference 4 get \(69*4 = 276\) and finally add the first term -32, giving \(a_{70} = -32 + 276 = 244\).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial component. It is the constant amount that each term in the sequence adds to the previous term to get to the following term. Understanding the common difference helps in quickly identifying how the sequence progresses.

To find the common difference in a sequence, subtract the first term from the second term. For example, if you have a sequence starting with -32 and then -28, the common difference is \( -28 - (-32) = 4 \).

Common differences can be positive, negative, or zero, depending on whether the sequence is increasing, decreasing, or constant:

• If the common difference is positive, the sequence increases.
• If it's negative, the sequence decreases.
• If it is zero, every term in the sequence is the same.

Understanding the common difference allows you to predict future terms in an arithmetic sequence easily.

First Term

The first term of an arithmetic sequence is simply the starting number of the sequence. It is denoted by \(a_1\).

In an arithmetic sequence equation, it establishes the baseline from which all other terms are derived. When solving problems involving arithmetic sequences, you will often see the first term as part of the formula that generates subsequent terms.

For example, in the sequence problem where \(a_1 = -32\), this -32 is not just the first term but serves as an anchor. From this number, consistent steps (determined by the common difference) will lead to other terms in the sequence. Knowing the first term is essential because it directly influences every other calculation for terms that follow in the sequence.

Even when the arithmetic sequence involves deep calculations, the first term remains a key reference point for understanding how everything else in the pattern fits together.

Nth Term

The \(n^{th}\) term in an arithmetic sequence represents any term in that sequence, identified as the \(n^{th}\) position. The formula for the \(n^{th}\) term \(a_n\) allows you to find the value of any term in the sequence without having to write out all the terms before it.

The formula is:
\[ a_n = a_1 + (n-1) imes d \]
This formula uses:

• \(a_1\), which is the first term of the sequence.
• \(n\), the position number of the term you wish to find.
• \(d\), the common difference.

For example, in our exercise finding \(a_{70}\), use the formula to calculate \(a_{70} = -32 + (70-1) imes 4 = 244\).

This structured approach is helpful, especially for large \(n\) values, allowing efficient calculation for terms far into the sequence without manual counting. Understanding this concept lets you deal with complex arithmetic problems more efficiently.