QuestionQuestion Show Write an explicit formula that represents the sequence defined by the following recursive formula: $$a_{1}=5 \text { and } a_{n}=-4 a_{n-1}$$
Answer Attempt 1 out of 2 $$a_{n}=$$ $\square$ Submit Answer

Studdy Solution
Prove by induction that $a_n = 5 \cdot (-4)^{n-1}$ satisfies the recursive relation $a_n = -4 a_{n-1}$ and the initial condition $a_1 = 5$.
**Base Case:** For $n = 1$,
$a_1 = 5 = 5 \cdot (-4)^{1-1} = 5 \cdot 1 = 5$
So, the base case holds.
**Induction Step:** Assume that $a_k = 5 \cdot (-4)^{k-1}$ holds for some $k \geq 1$. We need to show that $a_{k+1} = 5 \cdot (-4)^k$.
Using the recursive relation:
$a_{k+1} = -4 a_k = -4 \cdot 5 \cdot (-4)^{k-1} = 5 \cdot (-4)^k$
Hence, by the principle of mathematical induction, the explicit formula $a_n = 5 \cdot (-4)^{n-1}$ holds for all $n \geq 1$.
The explicit formula for the sequence is:
$a_n = 5 \cdot (-4)^{n-1}$