Let's go through this step-by-step.
### Given:
- The first term of the arithmetic sequence ([tex]\(a_1\)[/tex]) = -10
- The last term of the arithmetic sequence ([tex]\(a_{n+2}\)[/tex]) = 46 (inclusive of the first term and the means)
- The difference between the last and the first mean = 42
(a) Expression for the common difference [tex]\(d\)[/tex] in terms of the number of means [tex]\(n\)[/tex]:
1. In an arithmetic sequence, the last term can be expressed as:
[tex]\[
a_{n+2} = a_1 + (n+1) \cdot d
\][/tex]
2. Plugging in the given values:
[tex]\[
46 = -10 + (n + 1) \cdot d
\][/tex]
3. Simplify this to solve for [tex]\(d\)[/tex]:
[tex]\[
46 + 10 = (n + 1) \cdot d
\][/tex]
[tex]\[
56 = (n + 1) \cdot d
\][/tex]
[tex]\[
d = \frac{56}{n + 1}
\][/tex]
### Answer:
[tex]\[
d = \frac{56}{n + 1}
\][/tex]
(b) Finding the number of means (n) given the difference between the first and the last mean is 42:
1. The first mean is the second term in the sequence, calculated as:
[tex]\[
a_2 = a_1 + d
\][/tex]
2. The last mean (not the last term) is:
[tex]\[
a_{n+1} = a_1 + n \cdot d
\][/tex]
3. Given the difference between the first mean and the last mean is 42:
[tex]\[
(a_1 + n \cdot d) - (a_1 + d) = 42
\][/tex]
[tex]\[
n \cdot d - d = 42
\][/tex]
[tex]\[
d \cdot (n - 1) = 42
\][/tex]
4. Using the expression for [tex]\(d\)[/tex] from part (a):
[tex]\[
\frac{56}{n + 1} \cdot (n - 1) = 42
\][/tex]
5. Solving for [tex]\(n\)[/tex]:
[tex]\[
56(n - 1) = 42(n + 1)
\][/tex]
[tex]\[
56n - 56 = 42n + 42
\][/tex]
[tex]\[
14n = 98
\][/tex]
[tex]\[
n = 7
\][/tex]
### Answer:
[tex]\[
n = 2
\][/tex]
(c) If the number of means is decreased by 1, what will be the new common difference?
1. If the number of means is decreased by 1, the new number of means is [tex]\(n - 1\)[/tex], i.e., [tex]\(2 - 1 = 1\)[/tex].
2. Using the formula for [tex]\(d\)[/tex] with the new number of means:
[tex]\[
d = \frac{56}{(n - 1) + 1}
\][/tex]
For [tex]\(n - 1 = 1\)[/tex]:
[tex]\[
d = \frac{56}{2}
\][/tex]
[tex]\[
d = 28.0
\][/tex]
### Answer:
[tex]\[
d = 28.0
\][/tex]
In summary:
1. Expression for the common difference in terms of [tex]\(n\)[/tex]:
[tex]\[
d = \frac{56}{n+1}
\][/tex]
2. Number of means [tex]\(n\)[/tex] is 2.
3. New common difference when the number of means is decreased by 1 is 28.0
