MP3 player

Mary plans to buy a new MP3 player for 90 USD. And every month, the amount he saved in the previous month doubles. If she saves 3 USD in the first month, how many months will Mary save enough to buy an MP3 player?

Correct answer:

n =  5

Step-by-step explanation:

$$\begin{gathered} a_1=3\text{USD} \\ q=2 \\ s=90\text{USD} \\ s=a_1\cdot \frac{q^n-1}{q-1} \\ {q}^n=1+\frac{s\cdot (q-1)}{a_1} \\ k=1+\frac{s\cdot (q-1)}{a_1}=1+\frac{90\cdot (2-1)}{3}=31 \\ {q}^n=k \\ n\ln q=\ln k \\ {n}_1=\ln k/\ln q=\ln 31/\ln 2\doteq 4.9542 \\ n=[n_1]=[4.9542]=5 \end{gathered}$$

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