To find the missing terms in the arithmetic sequence where we have the known terms 27 and 53, we need to follow a few steps.
Step 1: Identify the given terms
We have the following terms in the sequence:
- First known term: 27
- Last known term: 53
Step 2: Determine the position of the terms
Let's denote the sequence in order:
- The 1st term: ...,
- The 2nd term: 27,
- The 3rd term: ?,
- The 4th term: ?,
- The 5th term: ?,
- The 6th term: 53,
- The 7th term: ...
Here, we need to find the 3rd, 4th, and 5th terms.
Step 3: Find the common difference
Since this is an arithmetic sequence, the difference between each term is constant. We can find this common difference by using the known terms.
Between the 2nd and the 6th term, there are \(6 - 2 = 4\) intervals:
\[ \text{Number of intervals} = 4 \]
The difference between the 2nd term (27) and the 6th term (53) is:
\[ 53 - 27 = 26 \]
Now, we can find the common difference \(d\):
\[ d = \frac{53 - 27}{4} = \frac{26}{4} = 6.5 \]
Step 4: Calculate the missing terms
Now that we know the common difference \(d = 6.5\), we can calculate the missing terms:
- 3rd term:
\[ \text{3rd term} = \text{2nd term} + d = 27 + 6.5 = 33.5 \]
- 4th term:
\[ \text{4th term} = \text{3rd term} + d = 33.5 + 6.5 = 40 \]
- 5th term:
\[ \text{5th term} = \text{4th term} + d = 40 + 6.5 = 46.5 \]
Step 5: List all the terms in the sequence
Now we can write the sequence with the calculated terms included:
..., 27, 33.5, 40, 46.5, 53, ...
Conclusion
The missing terms in the arithmetic sequence are:
- 3rd term: 33.5
- 4th term: 40
- 5th term: 46.5
So, the complete sequence including the missing terms is: ..., 27, 33.5, 40, 46.5, 53, ...