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2020-11-30

The Palmers and the Reynolds are saving money for a trip to Utah to go snowboarding. The Arnolds are going to save a nickel on the first day of the month and then double the amount each day for a month. The Gaineys are going to start their savings by saving $10 on the first day and then$10 each day of the month. Write a recursive and explicit formula for each option.

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The Reynolds save $0.05 on the first day, and then double the amount in their savings each day after that. Since the amount in their savings is increasing by a constant factor of 2, then the amount in their savings is represented by a geometric sequence with a constant ratio of r= 2 and first term of ${a}_{1}=0.05$. The recursive formula for a geometic sequence is ${a}_{n}=r×{a}_{n-1}$. We know that r= 2 so the recursive formula for the Reynolds is ${a}_{1}=0.05$, ${a}_{n}=2{a}_{n-1}$. The explicit formula for a geometric sequence is ${a}_{n}={a}_{1}{r}^{n-1}$. We know ${a}_{1}=0.05$ and r= 2 so the explicit formula for the Reynolds is then ${a}_{n}=0.05\left(2{\right)}^{n-1}$. The Palmers save$10 on the first day, and then add \$10 more dollars each day after that. Since the amount in their savings in increasing by a constant amount of 10, then the amount in their savings is represented by an arithmetic sequence with a common difference of d=10 and a first term of ${a}_{1}=10$.
The recursive formula forarithmetic sequence is ${a}_{n}={a}_{n-1}+d$. We know d=10 so the recursive formula for Palmers is then ${a}_{1}=10$, ${a}_{n}={a}_{n-1}+10$
The explicit formula for an arithmetic sequence is ${a}_{n}={a}_{1}+\left(n-1\right)d$. We know ${a}_{1}=10$ and d=10 so the explicit formula for the Palmers is then ${a}_{n}=10+\left(n-1\right)\left(10\right)$

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