Question

A sequence is defined by f(0)=3 , f(n)=2 f(n-1)

a.) Explain how you can tell the above definition is recursive.
b.) Write an explicit formula to represent f(n)
c.) Use the explicit formula to find the 15th term of the sequence.

Answer 1

a.) The given definition is recursive because it defines the function f(n) in terms of its previous value, f(n-1).

b.) The explicit formula for the sequence is
`\( f(n) = 3 \cdot 2^n \).`

c.) Using the explicit formula, the 15th term of the sequence is
`\( f(15) = 3 \cdot 2^(15) = 49152 \).`

Step-by-step explanation:

The provided definition of the sequence is recursive because it expresses each term, f(n), in terms of the preceding term, f(n-1). In the given case, f(0) is given as the initial value, and f(n) is defined as 2 times the value of f(n-1). This recursive nature is evident in the formula, where each term depends on the value of the previous term, creating a chain of dependencies.

To find an explicit formula, we observe the pattern in the recursive definition. The function f(n) is a geometric sequence with an initial term of 3 and a common ratio of 2. The general formula for a geometric sequence is
`\( a_n = a_0 \cdot r^n \), where \( a_n \) is the nth term, \( a_0 \)` is the initial term, r is the common ratio, and n is the term number. Applying this formula to the given sequence, we get
`\( f(n) = 3 \cdot 2^n \).`

Using the explicit formula, we can easily find the 15th term of the sequence by plugging in n=15 into the formula:
`\( f(15) = 3 \cdot 2^(15) = 49152 \).`This provides a direct and efficient way to compute the value of any term in the sequence without having to recursively calculate each preceding term.