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
\
