Question

In Problems \(31-38\), find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 8th term is \(8 ; 20\) th term is 44

Short answer

First term = -13; common difference = 3; recursive formula: a_n = a_{n-1} + 3; nth term formula: a_n = 3n - 16.

Step-by-step solution

Identify the given terms

The 8th term of the sequence is 8, and the 20th term of the sequence is 44. These will be denoted as follows: a_8 = 8 a_{20} = 44.

Setup the general formula for an arithmetic sequence

The general formula for the nth term of an arithmetic sequence is given by a_n = a_1 + (n-1)d where a_n is the nth term, a_1 is the first term, and d is the common difference.

Setup the equations based on the given terms

Using the general formula: a_8 = a_1 + 7d = 8 a_{20} = a_1 + 19d = 44.

Solve the system of equations

We have two equations to solve: a_1 + 7d = 8 a_1 + 19d = 44 Subtract the first equation from the second to eliminate a_1: (a_1 + 19d) - (a_1 + 7d) = 44 - 8 12d = 36 Solving for d: d = 36/12 = 3.

Find the first term

Using the value of d in the first equation: a_1 + 7(3) = 8 a_1 + 21 = 8 a_1 = 8 - 21 a_1 = -13.

Write the recursive formula for the sequence

The recursive formula for an arithmetic sequence is given by a_{n} = a_{n-1} + d. Therefore, the recursive formula is a_n = a_{n-1} + 3 with a_1 = -13.

Write the formula for the nth term

Substitute a_1 and d into the nth term formula: a_n = -13 + (n-1)3. Simplify: a_n = -13 + 3n - 3 a_n = 3n - 16.

Key concepts

nth term formula

To understand how to find any specific term in an arithmetic sequence, we use the nth term formula. This formula helps us determine the value of any term based on its position in the sequence. The nth term formula for an arithmetic sequence is:

\[ a_n = a_1 + (n-1)d \]

In this formula,

• \(a_n\) is the term we want to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference between the terms.

For example, if the first term \(a_1\) is -13 and the common difference \(d\) is 3, the nth term can be calculated as follows:

\[ a_n = -13 + (n-1) \times 3 \]
By simplifying further, we get:

\[a_n = -13 + 3n - 3 = 3n - 16\]
Therefore, the general formula for the nth term of this sequence is \(a_n = 3n - 16\). This formula is very useful for quickly finding any term in the sequence without having to list all the terms.

recursive formula

A recursive formula is another way to express the terms of a sequence. Instead of finding a term directly like in the nth term formula, it finds a term based on the previous term. For an arithmetic sequence, the recursive formula is:

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

• \(a_n\) is the current term.
• \(a_{n-1}\) is the previous term.
• \(d\) is the common difference.

This formula shows that each term in the sequence is equal to the previous term plus the common difference. For our specific example, where \(a_1 = -13\) and \(d = 3\), the recursive formula is:

\[ a_n = a_{n-1} + 3 \]
Using this recursive formula:

• To find \(a_2\), you add 3 to \(a_1\):
\( a_2 = -13 + 3 = -10 \)
• To find \(a_3\), you add 3 to \(a_2\):
\( a_3 = -10 + 3 = -7 \)

And this process continues for finding further terms. Recursive formulas are particularly useful for sequences where each term depends on its predecessor.

common difference

The common difference is a key feature of an arithmetic sequence. It is the fixed amount added to each term to get the next term. In our example, the common difference \(d\) was found by solving a system of equations. Specifically:

1. Given two terms: \(a_8 = 8\) and \(a_{20} = 44\)
2. Use the nth term formula: \( a_n = a_1 + (n-1)d \)
3. Create two equations:
- \( a_8 = a_1 + 7d = 8 \)
- \( a_{20} = a_1 + 19d = 44 \)
4. Subtract the first equation from the second to eliminate \(a_1\):
-\( 12d = 36 \), thus \( d = 3 \)

This common difference remains the same throughout the sequence. For instance, from our equations:

• \( -13 + 7 \times 3 = 8 \)
• \( -13 + 19 \times 3 = 44 \)

This confirms the common difference \(d = 3\) is correct.

system of equations

A system of equations can be used to solve for unknowns within the context of an arithmetic sequence. For our example, we used a system of equations to find the first term \(a_1\) and the common difference \(d\). Here is a step-by-step breakdown:

1. Write down the information from the problem:

• \(a_8 = 8\)
• \(a_{20} = 44\)

2. Use the nth term formula to create two equations:
\( a_8 = a_1 + 7d \)
\( a_{20} = a_1 + 19d \)

3. Simplify and combine the equations:

• \( a_1 + 7d = 8 \)
• \( a_1 + 19d = 44 \)

4. Subtract one equation from the other to eliminate \(a_1\):
\( 12d = 36 \)
\( d = 3 \)

5. Substitute \(d\) back into one of the original equations to find \(a_1\):

• \( a_1 + 21 = 8 \)
• \( a_1 = -13 \)

This method is invaluable when dealing with sequences or any scenario where multiple variables need to be solved at once. By understanding and setting up a system of equations, we can unravel complex problems into manageable steps.