Question

The first two numbers in a sequence h are h(1)=2 and h(2)=6.

Part a:

If h is an arithmetic sequence, write a definition for the nth term of h. Explain your reasoning.

Part b:

If h is a geometric sequence, write a definition for the nth term of h. Explain your reasoning.

Answer 1

If h is an arithmetic sequence, the nth term can be defined as h(n) = h(1) + (n-1)d. If h is a geometric sequence, the nth term can be defined as h(n) = h(1) * r^(n-1).

Step-by-step explanation:

Part a: If the sequence h is an arithmetic sequence, the nth term can be defined using the formula:

h(n) = h(1) + (n-1)d

where h(1) is the first term, n is the position of the term in the sequence, and d is the common difference between consecutive terms. In this case, h(1) = 2 and h(2) = 6, so the common difference is d = h(2) - h(1) = 6 - 2 = 4. Therefore, the nth term of h is given by:

h(n) = 2 + 4(n-1)

Part b: If the sequence h is a geometric sequence, the nth term can be defined using the formula:

h(n) = h(1) * r^(n-1)

where h(1) is the first term, n is the position of the term in the sequence, and r is the common ratio between consecutive terms. In this case, h(1) = 2 and h(2) = 6, so the common ratio is r = h(2) / h(1) = 6 / 2 = 3. Therefore, the nth term of h is given by:

h(n) = 2 * 3^(n-1)