Arithmetic Sequences

Instruction

Finding the Next Term of an Arithmetic Sequence

Consider the arithmetic sequence [tex]27, 40, 53, 66, 79, \ldots[/tex]

The next two terms are [tex]\square[/tex] and [tex]\square[/tex].

What rule can be used to find the next term of the sequence?

[tex]
\begin{aligned}
a_n & = a_{n-1} - 79 \\
a_n & = a_{n-1} - 13 \\
a_n & = a_{n-1} + 13 \\
a_n & = a_{n-1} + 79
\end{aligned}
[/tex]



Answer :

To solve the problem of finding the next two terms of the given arithmetic sequence and determining the rule for the sequence, we'll proceed with the following steps:

1. Identify the Sequence:
The given sequence is: \( 27, 40, 53, 66, 79, \ldots \).

2. Determine the Common Difference:
In an arithmetic sequence, the difference between any two consecutive terms is constant. This difference is called the common difference. We find it by subtracting the first term from the second term:
[tex]\[ 40 - 27 = 13 \][/tex]
So, the common difference \( d \) is 13.

3. Find the Next Two Terms:
To find the next term after 79, we add the common difference to the last known term.
[tex]\[ 79 + 13 = 92 \][/tex]
Therefore, the next term is 92.

To find the following term, we add the common difference again to the term we just found:
[tex]\[ 92 + 13 = 105 \][/tex]
Therefore, the term after 92 is 105.

The next two terms of the sequence are 92 and 105.

4. Determine the Rule for the Sequence:
In an arithmetic sequence, each term \(a_n\) can be found by adding the common difference to the previous term \(a_{n-1}\). Therefore, the rule for finding the terms in this sequence is:
[tex]\[ a_n = a_{n-1} + 13 \][/tex]

Collecting our findings:
- The common difference \( d \) is 13.
- The next two terms are 92 and 105.
- The rule for the sequence is \( a_n = a_{n-1} + 13 \).

Thus, the complete solution is:
1. The next two terms are 92 and 105.
2. The rule to find the next term of the sequence is:
\[
a_n = a_{n-1} + 13
\