Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Insert three arithmetic means between 3 and \(-5\).

Step-by-Step Solution

Answer

The arithmetic means are 1, -1, and -3.

Step 1: Understand Arithmetic Sequence

An arithmetic sequence, or arithmetic progression, is a sequence of numbers where the difference between any two consecutive terms is constant. This difference is called the common difference.

Step 2: Identify the First and Last Terms

We are given the first term as 3 and the last term as -5, and we need to find three arithmetic means between them. This means our sequence has 5 terms: 3, A1, A2, A3, -5.

Step 3: Determine Common Difference

Let the common difference be denoted by \(d\). The nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\). Here, the 5th term is -5, and can be found by \(-5 = 3 + 4d\).

Step 4: Solve for the Common Difference

Rearrange \(-5 = 3 + 4d\) to solve for \(d\). We get \(4d = -5 - 3\) which simplifies to \(4d = -8\). Hence, \(d = -2\).

Step 5: Insert Arithmetic Means

Now that we have the common difference \(d = -2\), we can insert the arithmetic means: A1, A2, and A3. Calculate each term with the formula \(a_n = a_1 + (n-1)d\):- A1 = \(3 + 1(-2) = 1\)- A2 = \(3 + 2(-2) = -1\)- A3 = \(3 + 3(-2) = -3\).

Step 6: Verify the Sequence

Verify that the entire sequence with the means inserted is correct: 3, 1, -1, -3, -5. Each term should differ from the next by a common difference of -2, which matches perfectly.

Key Concepts

Common Difference

When exploring arithmetic sequences, a key element to understand is the **common difference**. It's the consistent interval between consecutive terms in the sequence.
In any arithmetic sequence, each term can be found by adding this common difference to the previous term.
For example, if we have an arithmetic sequence like 3, 5, 7, 9, the common difference here is 2, because 5 - 3 = 2, 7 - 5 = 2, and so forth.
Generally speaking, if you denote the common difference by \(d\), then every term after the first can be expressed as:
\[ a_n = a_1 + (n-1)d \]
Here, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. This formula helps compute any term in the sequence by understanding the common difference.

Arithmetic Means

**Arithmetic means** are additional terms inserted into an arithmetic sequence to extend it or fill in gaps between specified terms.
Inserting arithmetic means between two numbers creates a longer sequence while maintaining consistent intervals (the common difference) between all terms.
To find arithmetic means between two numbers, follow these steps:

• Identify the starting and ending terms of the sequence.
• Determine how many means need to be inserted.
• Calculate the common difference \(d\) using the formula for the nth term of an arithmetic sequence, and solve for \(d\).

Once you have the common difference, use it to calculate each of the arithmetic means by applying the sequence formula \( a_n = a_1 + (n-1)d \). Each mean gets inserted into the sequence to create a uniform progression.

Sequence of Numbers

A **sequence of numbers** is essentially a collection of numbers arranged in a specific order, according to some defined rule or pattern.
An arithmetic sequence is a prime example, where the numbers form a pattern by having the same common difference.
How to recognize or construct an arithmetic sequence?

• Check if the difference between consecutive numbers remains constant. This confirms it's arithmetic.
• The number of terms, the first term, and the common difference are key aspects in defining the sequence.

For example, given the sequence 3, 1, -1, -3, -5, the common difference \(-2\) confirms it's arithmetic, as each number is decremented by 2 from its predecessor.
Practicing with these sequences helps in understanding the systematic progression of numbers and aids in solving related problems efficiently.