Answer :
Certainly! Let's solve the problem step-by-step.
Question:
- Given the first term [tex]\( a = -4 \)[/tex] and the last term [tex]\( b = 8 \)[/tex].
- We need to find the arithmetic mean(s) between these terms.
Solution:
1. Understand the Problem:
- An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference and is denoted as [tex]\( d \)[/tex].
- Our goal is to insert one arithmetic mean between the given terms [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
2. Set Up the Sequence:
- The given terms are:
- First term ([tex]\(a\)[/tex]) = -4
- Last term ([tex]\(b\)[/tex]) = 8
- Number of means to insert: 1
3. Determine the Common Difference:
- The sequence will be of the form: [tex]\( a, M, b \)[/tex]
- Here, [tex]\( M \)[/tex] is the arithmetic mean we need to find.
- Since there is only one mean to insert, the sequence consists of 3 terms: [tex]\( a \)[/tex], [tex]\( M \)[/tex], and [tex]\( b \)[/tex].
- We can use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence: [tex]\( a + n \cdot d \)[/tex], where [tex]\( n \)[/tex] is the position of the term in the sequence and [tex]\( d \)[/tex] is the common difference.
- For [tex]\( a \)[/tex] (the first term) and [tex]\( b \)[/tex] (the third term), we have:
[tex]\[ b = a + 2d \][/tex]
Since [tex]\( b \)[/tex] is the third term (corresponding to [tex]\( n = 2 \)[/tex]).
4. Plug in the Given Values:
- [tex]\( a = -4 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- Substitute these values into the equation:
[tex]\[ 8 = -4 + 2d \][/tex]
5. Solve for [tex]\( d \)[/tex]:
- Rearrange the equation to solve for [tex]\( d \)[/tex]:
[tex]\[ 8 + 4 = 2d \][/tex]
[tex]\[ 12 = 2d \][/tex]
[tex]\[ d = \frac{12}{2} = 6 \][/tex]
6. Calculate the Arithmetic Mean:
- Since we need to find the mean [tex]\( M \)[/tex], which is the second term in the sequence:
[tex]\[ M = a + d \][/tex]
[tex]\[ M = -4 + 6 = 2 \][/tex]
Conclusion:
- The common difference [tex]\( d \)[/tex] is [tex]\( 6 \)[/tex].
- The arithmetic mean between [tex]\(-4\)[/tex] and [tex]\(8\)[/tex] is [tex]\(2\)[/tex].
So, the step-by-step solution shows that the common difference is [tex]\( 6 \)[/tex] and the arithmetic mean is [tex]\( 2 \)[/tex].
Question:
- Given the first term [tex]\( a = -4 \)[/tex] and the last term [tex]\( b = 8 \)[/tex].
- We need to find the arithmetic mean(s) between these terms.
Solution:
1. Understand the Problem:
- An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference and is denoted as [tex]\( d \)[/tex].
- Our goal is to insert one arithmetic mean between the given terms [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
2. Set Up the Sequence:
- The given terms are:
- First term ([tex]\(a\)[/tex]) = -4
- Last term ([tex]\(b\)[/tex]) = 8
- Number of means to insert: 1
3. Determine the Common Difference:
- The sequence will be of the form: [tex]\( a, M, b \)[/tex]
- Here, [tex]\( M \)[/tex] is the arithmetic mean we need to find.
- Since there is only one mean to insert, the sequence consists of 3 terms: [tex]\( a \)[/tex], [tex]\( M \)[/tex], and [tex]\( b \)[/tex].
- We can use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence: [tex]\( a + n \cdot d \)[/tex], where [tex]\( n \)[/tex] is the position of the term in the sequence and [tex]\( d \)[/tex] is the common difference.
- For [tex]\( a \)[/tex] (the first term) and [tex]\( b \)[/tex] (the third term), we have:
[tex]\[ b = a + 2d \][/tex]
Since [tex]\( b \)[/tex] is the third term (corresponding to [tex]\( n = 2 \)[/tex]).
4. Plug in the Given Values:
- [tex]\( a = -4 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- Substitute these values into the equation:
[tex]\[ 8 = -4 + 2d \][/tex]
5. Solve for [tex]\( d \)[/tex]:
- Rearrange the equation to solve for [tex]\( d \)[/tex]:
[tex]\[ 8 + 4 = 2d \][/tex]
[tex]\[ 12 = 2d \][/tex]
[tex]\[ d = \frac{12}{2} = 6 \][/tex]
6. Calculate the Arithmetic Mean:
- Since we need to find the mean [tex]\( M \)[/tex], which is the second term in the sequence:
[tex]\[ M = a + d \][/tex]
[tex]\[ M = -4 + 6 = 2 \][/tex]
Conclusion:
- The common difference [tex]\( d \)[/tex] is [tex]\( 6 \)[/tex].
- The arithmetic mean between [tex]\(-4\)[/tex] and [tex]\(8\)[/tex] is [tex]\(2\)[/tex].
So, the step-by-step solution shows that the common difference is [tex]\( 6 \)[/tex] and the arithmetic mean is [tex]\( 2 \)[/tex].
