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 a1=0.05.  The recursive formula for a geometic sequence is an=r×an−1. We know that r= 2 so the recursive formula for the Reynolds is a1=0.05, an=2an−1.  The explicit formula for a geometric sequence is an=a1rn−1. We know a1=0.05 and r= 2 so the explicit formula for the Reynolds is then an=0.05(2)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 a1=10.  The recursive formula forarithmetic sequence is an=an−1+d. We know d=10 so the recursive formula for Palmers is then a1=10, an=an−1+10  The explicit formula for an arithmetic sequence is an=a1+(n−1)d. We know a1=10 and d=10 so the explicit formula for the Palmers is then an=10+(n−1)(10)

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 mo

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