Questions: A geometric sequence, g(n) starts 20, 60, ... Define g recursively and for the nth term.

Solution Steps

To define the geometric sequence recursively, we need to identify the first term and the common ratio. The sequence starts with 20 and 60, so the first term \( g(0) \) is 20. The common ratio can be found by dividing the second term by the first term, which is \( \frac{60}{20} = 3 \). Thus, the recursive definition is \( g(n) = 3 \times g(n-1) \) for \( n \geq 1 \). For the explicit formula, the \( n^{th} \) term of a geometric sequence is given by \( g(n) = g(0) \times r^n \), where \( r \) is the common ratio.

The geometric sequence starts with the terms 20 and 60. The first term is \( g(0) = 20 \). The common ratio \( r \) is calculated by dividing the second term by the first term: \[ r = \frac{60}{20} = 3 \]

The recursive formula for a geometric sequence is given by: \[ g(n) = r \times g(n-1) \] Substituting the known values, we have: \[ g(n) = 3 \times g(n-1) \] with the initial condition: \[ g(0) = 20 \]

The explicit formula for the \( n^{th} \) term of a geometric sequence is: \[ g(n) = g(0) \times r^n \] Substituting the known values, we have: \[ g(n) = 20 \times 3^n \]

Using the explicit formula, calculate the \( 5^{th} \) term: \[ g(5) = 20 \times 3^5 = 20 \times 243 = 4860 \]

Final Answer