Problem 30 - College Algebra

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=-70, d=-5$$

Step-by-Step Solution

Answer

The general term of the sequence is given by the formula: \(a_{n} = -5n -65\). The 20th term of the sequence \(a_{20} = -165\).

Step 1: Writing a general formula for `nth` term of an arithmetic sequence

In arithmetic sequences, any term `a_{n}` can be computed using the formula \(a_{n}=a_{1}+ (n-1)*d\). In this case, \(a_{1}=-70\) and \(d=-5\). Therefore, the formula becomes \(a_{n}= -70 + (n-1)(-5)\).

Step 2: Simplify the formula

Distribute the multiplication in the parenthesis, the formula simplifies to \(a_{n}=-70 - 5n + 5 = -5n - 65\).

Step 3: Finding the 20th term.

Now, use this formula to find \(a_{20}\). Substitute 20 for `n` in the formula, \(a_{20}=-5(20)-65 = -100 - 65 = -165\). Thus, the 20th term of the sequence is -165.

Key Concepts

General Term Formula

In arithmetic sequences, a general term formula allows us to determine any term in the sequence without having to list all previous terms. This formula is crucial for understanding patterns and predicting future terms. For an arithmetic sequence, the general term, often denoted as \(a_n\), is determined by the formula: \[a_{n} = a_{1} + (n-1) imes d\] where:

• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference between consecutive terms.
• \(n\) represents the position of the term in the sequence.

This formula takes the initial term and adds the common difference multiplied by the number of steps you need to reach the \(n^{th}\) term from the first term. It simplifies the process of finding any term in the sequence.

nth Term

The "nth term" is a concept used to describe a particular term in the sequence at an arbitrary position \(n\). It represents the value of the sequence at any chosen point. In our context, calculating the nth term efficiently is about applying the general term formula. For example, if we want to compute the 20th term of a sequence starting at \(a_1 = -70\) with a common difference \(d = -5\), you insert \(n=20\) into the simplified formula: \[a_{20} = -5n - 65\] Calculating \(a_{20}\) by substituting \(n = 20\), gives us \(-5(20) - 65 = -165\). This direct method speeds up the determination of any term without sequential addition.

Common Difference

The common difference is a key feature of an arithmetic sequence. It is the difference between consecutive terms and remains constant throughout the sequence. In other words, it's the "gap" between one term and the next, ensuring predictability in the sequence. To find the common difference \(d\), subtract any term from the subsequent term.

• For instance, if \(a_2 = a_1 + d\), then \(d = a_2 - a_1\).

In our example, the common difference \(d\) is \(-5\). Whenever you move along the sequence, you subtract 5 to get the next term, creating a linear, predictable pattern.

Sequence

A sequence is essentially a set of numbers arranged in a specific mathematical order. An arithmetic sequence is a type where the difference between any two consecutive terms is constant. These sequences are simple but powerful tools in mathematics, as they represent evenly spread numbers over a particular interval. Each term's position is important as it dictates its value, derived from the general formula. For instance, in the given sequence starting with \(a_1 = -70\) and \(d = -5\), the sequence unfolds as follows:

• \(a_1 = -70\),
• \(a_2 = -75\),
• \(a_3 = -80\), ...

The terms continue progressively with the fixed amount \(-5\) subtracted each time. Understanding sequences helps in interpreting trends, projections, and many practical applications.