Question

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

a. If he is an arithmetic sequence, write a definition for the nth term of h. Explain your reasoning.
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b. If h is a geometric sequence, write a definition for the nth term of h. Explain your reasoning.
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Answer 1

The nth term of the arithmetic sequence h can be defined as h(n) = 2 + (n-1)4, and the nth term of the geometric sequence h can be defined as h(n) = 2 * 3^(n-1).

Step-by-step explanation:

a. If the sequence h is an arithmetic sequence, the nth term can be defined using the formula: h(n) = a + (n-1)d, where 'a' is the first term and 'd' is the common difference. In this case, h(1) = 2 and h(2) = 6. Using these values, we can find the common difference (d) by subtracting the second term from the first term: d = h(2) - h(1) = 6 - 2 = 4. Therefore, the nth term of h is h(n) = 2 + (n-1)4.

b. If the sequence h is a geometric sequence, the nth term can be defined using the formula: h(n) = ar^(n-1), where 'a' is the first term and 'r' is the common ratio. In this case, we need to determine if the sequence h is a geometric sequence by checking if the ratio of consecutive terms is constant. Let's calculate the ratio: r = h(2)/h(1) = 6/2 = 3. Since the ratio is constant, the sequence h is a geometric sequence, and the nth term is h(n) = 2 * 3^(n-1).

Learn more about nth term of sequences