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Answer:

Step-by-step explanation:

Let's solve the given arithmetic sequence problem step by step:

a) To find the common difference (d), we can use the formula for the nth term of an arithmetic sequence:

An = A1 + (n - 1)d

where An is the nth term, A1 is the first term, and d is the common difference.

Given that the 5th term (A5) is 28 and the 9th term (A9) is 40, we can write the following equations:

A5 = A1 + 4d   ...(1)
A9 = A1 + 8d   ...(2)

Substituting the given values:
28 = A1 + 4d   ...(1)
40 = A1 + 8d   ...(2)

To eliminate A1, subtract equation (1) from equation (2):
40 - 28 = A1 + 8d - (A1 + 4d)
12 = 4d
d = 3

Therefore, the common difference (d) of the arithmetic sequence is 3.

b) To find the 13th term (A13), we use the formula:

A13 = A1 + (13 - 1)d
A13 = A1 + 12d
A13 = A1 + 12(3)
A13 = A1 + 36

To find the first term (A1), we can substitute the common difference (d) into equation (1):

28 = A1 + 4(3)
28 = A1 + 12
A1 = 16

Therefore, the 13th term (A13) is A1 + 36 = 16 + 36 = 52, and the first term (A1) is 16.

c) To determine if 100 is a difference of any two terms in this sequence, we can check if there exists an integer n such that:

An - Am = 100

Substituting the values into the formula:

(A1 + (n - 1)d) - (A1 + (m - 1)d) = 100

Simplifying:

(n - m)d = 100

Since d = 3 (the common difference), we have:

3(n - m) = 100

However, there is no integer value for n and m that satisfies this equation. Therefore, 100 is not the difference of any two terms in this sequence.

d) The algebraic form of this arithmetic sequence can be written as:

An = A1 + (n - 1)d

Substituting the values we found earlier:

An = 16 + (n - 1)3

Therefore, the algebraic form of this sequence is An = 16 + 3n - 3.

Note: In the final equation, the term (3n - 3) can be further simplified or rearranged if desired.

Answer:

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Step-by-step explanation:

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