Question

Consider the sequence 3, 6, 12, 24, 48, .... (a) Write a recursive rule to represent the sequence. (b) Write an explicit rule to represent the sequence. (c) Find the 15th term in the sequence.

Answer 1

Answer:

  a. a[1] = 3; a[n] = 2a[n-1]

  b. a[n] = 3·2^(n-1)

  c. a[15] = 49,152

Step-by-step explanation:

Each term of the given sequence is 2 times the previous term. (This description is the basis of the recursive formula.) That is, the terms of the given sequence have a common ratio of 2. This means the sequence is geometric, so the formulas for explicit and recursive rules for a geometric sequence apply.

The first term is 3, and the common ratio is 2.

(a)

The recursive rule is ...

  a[1] = 3

  a[n] = 2×a[n-1]

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(b)

The explicit rule is ...

  a[n] = a[1]×r^(n-1)

  a[n] = 3×2^(n-1)

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(c)

The 15th term is ...

  a[15] = 3×2^(15-1) = 3×2^14

  a[15] = 49,152