## Related questions with answers

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

VerifiedStep 1

1 of 2The 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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