Questions: Given the explicit formula below, write the recursive formula for the sequence. a(n)=11(-4)^(n-1)

Solution Steps

To convert the given explicit formula to a recursive formula, we need to express the \( n \)-th term of the sequence in terms of the previous term(s).

For the given explicit formula \( a(n) = 11(-4)^{n-1} \):

1. Identify the first term \( a(1) \).
2. Determine the common ratio or difference between consecutive terms.
3. Formulate the recursive relation using the identified first term and the common ratio or difference.

1. The first term \( a(1) \) can be found by substituting \( n = 1 \) into the explicit formula.
2. The common ratio \( r \) can be found by dividing \( a(n+1) \) by \( a(n) \).
3. The recursive formula will be of the form \( a(n) = r \cdot a(n-1) \) with the initial condition \( a(1) \).

To find the first term \( a(1) \) of the sequence, we substitute \( n = 1 \) into the explicit formula: \[ a(1) = 11(-4)^{1-1} = 11(-4)^0 = 11 \cdot 1 = 11 \]

The common ratio \( r \) can be found by dividing \( a(n+1) \) by \( a(n) \). Using the explicit formula: \[ a(n+1) = 11(-4)^{(n+1)-1} = 11(-4)^n \] \[ a(n) = 11(-4)^{n-1} \] \[ r = \frac{a(n+1)}{a(n)} = \frac{11(-4)^n}{11(-4)^{n-1}} = (-4) \]

Using the first term \( a(1) = 11 \) and the common ratio \( r = -4 \), the recursive formula can be written as: \[ a(n) = -4 \cdot a(n-1) \] with the initial condition: \[ a(1) = 11 \]

To verify the 5-th term, we use the recursive formula: \[ a(2) = -4 \cdot a(1) = -4 \cdot 11 = -44 \] \[ a(3) = -4 \cdot a(2) = -4 \cdot (-44) = 176 \] \[ a(4) = -4 \cdot a(3) = -4 \cdot 176 = -704 \] \[ a(5) = -4 \cdot a(4) = -4 \cdot (-704) = 2816 \]

Final Answer