Understanding Arithmetic Means
To find the 4 arithmetic means between -2 and 23, we start by recognizing that an arithmetic sequence is defined by a constant difference between consecutive terms.
Identifying the Sequence
1. First and Last Terms:
- The first term (a) is -2.
- The last term (l) is 23.
2. Total Number of Terms:
- We need to find 4 means, resulting in a total of 6 terms (2 endpoints + 4 means).
Finding the Common Difference
1. Arithmetic Series Formula:
- The formula for the nth term of an arithmetic sequence is given by:
- nth term = a + (n-1)d
- Here, 'd' is the common difference.
2. Calculate Common Difference:
- Using the formula for the last term:
- 23 = -2 + (6-1)d
- 23 = -2 + 5d
- 25 = 5d
- d = 5
Calculating the Arithmetic Means
1. List All Terms:
- First term: -2
- Second term: -2 + 5 = 3
- Third term: 3 + 5 = 8
- Fourth term: 8 + 5 = 13
- Fifth term: 13 + 5 = 18
- Sixth term: 23
2. Arithmetic Means:
- The 4 arithmetic means are:
- 3, 8, 13, 18
Conclusion
The four arithmetic means between -2 and 23 are 3, 8, 13, and 18. This sequence maintains a constant difference of 5 between consecutive terms, forming a complete arithmetic series.