Problem 30 - Algebra and Trigonometry

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 for the arithmetic sequence is \(a_{n} = -5n - 65\). The 20th term of the sequence is -165.

Step 1: Write the General Formula for Arithmetic Sequence

The general term of an arithmetic sequence can be written as follows: \(a_{n} = a_{1} + (n - 1) * d\). Here \(a_{n}\) is the nth term, \(a_{1}\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Step 2: Substitute the Given Values into the General Formula

Substitute the given values into the general formula to get \(a_{n} = -70 + (n - 1) * (-5)\). This can be simplified to \(a_{n} = -70 - 5n + 5\), and further simplified to \(a_{n} = -5n - 65\). This is the general term for the arithmetic sequence.

Step 3: Substitute n = 20 into the General Formula to get the 20th Term

To find the 20th term of the sequence, substitute \(n = 20\) into the general formula to get \(a_{20} = -5 * 20 - 65 = -100 - 65 = -165\).

Key Concepts

General Term

In every arithmetic sequence, you will come across the term called the "general term." This term represents a formula that can be used to find any specific term in a sequence without having to list all the terms before it. You could think of it as a shortcut to finding terms
Let’s break down the formula for the general term. The general term formula for any arithmetic sequence is:

\[a_{n} = a_{1} + (n - 1) imes d\]

- Here, \(a_{n}\) is the nth term you’re trying to find.- \(a_{1}\) is the first term in the sequence.- \(d\) is the common difference between terms.- \(n\) is the position of the term within the sequence.

Using this formula, you can find any term you need, knowing just the first term and the common difference.
This is extremely helpful because it allows you to skip all the intermediate terms.

Common Difference

A vital part of understanding an arithmetic sequence is recognizing the common difference. This difference determines how much each term in the sequence increases or decreases from the previous one.
Imagine you have marbles in a line, each one has more or fewer than the last by a certain fixed number. This number is your common difference.

To find the common difference:- Choose any term and subtract the previous term from it.

For example, with terms -70, -75, -80, -85:
\(-75 - (-70) = -5\), giving us a common difference of \(-5\).
You will use this common difference in the general term formula to easily calculate any term you need in the sequence.

Sequence Formula

The sequence formula in the context of arithmetic sequences refers to the general formula that defines the sequence of numbers. It sums up exactly how the numbers in the sequence are related.Using the general formula \(a_{n} = a_{1} + (n - 1) imes d\), you can create a sequence where each number is just a fixed number away from the one before it.
This allows you to deduce a pattern and predict future numbers in the sequence without effort.
To make use of the sequence formula:- Plug in the known values (first term and common difference).- Adjust for any position \(n\) you need.

If your first term is \(-70\) and your common difference \(-5\), your updated formula becomes \(a_{n} = -70 - (n - 1) imes 5\).This tells the story of your sequence, reflecting its path precisely.

Nth Term

The "nth term" refers to finding any particular term in the sequence, signified by \(n\). Often, this is the main goal when dealing with arithmetic sequences.

Let's say you're investigating the 20th term (denoted as \(a_{20}\)).To find this:- Use the general formula with \(n=20\).- Here's how it looks: \(a_{20} = -70 + (20 - 1) imes (-5)\).

After simplifying, we have \(a_{20} = -5 \times 20 - 65 = -165\).
This simple plug-in calculation demonstrates the power of the sequence formula.It allows you to accurately find any single term without computing all others sequentially.Arming yourself with the formula for the nth term of an arithmetic sequence means you're equipped to solve these problems swiftly and efficiently.