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Question

# A government program that currently costs taxpayers \$2.5 billion per year is cut back by 20 percent per year. Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.

Solution

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The sequence $a_n = 2.5\cdot10^9\cdot\left(\frac{8}{10}\right)^n$ is bounded below by $0$ and is a decreasing sequence. Theorem 8.5 states that $(a_n)$ is a $\text{\color{#4257b2} convergent sequence}$.

Lets find its limit:

\begin{align*} \color{#4257b2} \lim_{n \to \infty} a_n &=\lim_{n \to \infty}2.5\cdot10^9\cdot\left(\frac{8}{10}\right)^n \\ &=2.5\cdot10^9\cdot \lim_{n \to \infty}\left(\frac{8}{10}\right)^n \\ &= 2.5\cdot10^9\cdot 0 \\ &= \color{#4257b2} 0 \end{align*}

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