Concept explainers
Monthly Savings Program Alice opens a savings account that pays 3% interest per year, compounded monthly. She begins by depositing $100 at the start of the first month and adds $100 at the end of each month, when the interest is credited.
- (a) Find a recursive formula for the amount An in her account at the end of the nth month. (Include the interest credited for that month and her monthly deposit.)
- (b) Find the first five terms of the sequence An.
- (c) Use the pattern you observed in (b) to find a formula for An. [Hint: To find the pattern most easily, it’s best not to simplify the terms too much]
- (d) How much has she saved after 5 years?
(a)
The recursive formula for the amount
Answer to Problem 3P
The recursive formula for the amount
Explanation of Solution
Given:
Alice deposit $100 at the start of first month and then add $100 at the end of each month. The interest rate is 3% per year and interest compounded monthly.
Formula used:
The compound interest formula in given below,
Where, A is total amount after t years, r is the rate of interest per year and n is no of times interest is compounded per year
Calculation:
According to the given information the value of r is 3% or 0.03 and the value of n is 12. Alice add $100 at the end of the each month, so the amount after one month is calculated as follows,
Substitute 0.03 for r, 12 for n, 100 for P and
Add 100 and the total amount in first month after compounding the interest, to find the total amount after first month.
The new principle amount is
Substitute 0.03 for r, 12 for n,
Add 100 and the total amount in second month after compounding the interest, to find the total amount after second month.
The new principle amount is
Similarly, the principle amount after n month is
Therefore, the recursive formula for the amount
(b)
The first five terms of the sequence
Answer to Problem 3P
The first five terms of the sequence
Explanation of Solution
Given:
From part (a), the value of
Calculation:
The initial amount is $100. The value of first term
Substitute 1 for n in equation (2), to find the value of
Simplify the above expression to find the value of
Substitute 2 for n in equation (2), to find the value of
Simplify the above expression to find the value of
Substitute 3 for n in equation (2), to find the value of
Simplify the above expression to find the value of
Substitute 4 for n in equation (2), to find the value of
Simplify the above expression to find the value of
Therefore, the first five terms of the sequence
(c)
The formula for
Answer to Problem 3P
The formula for
Explanation of Solution
Given:
The first five terms of the sequence
Calculation:
The first five terms represents as shown below,
The second term is written as,
The third term is written as,
The fourth term is written as,
The fifth term is written as,
Similarly the general formula for
Therefore, the formula for
(d)
The total amount saved by Alice in 5 years.
Answer to Problem 3P
The total amount saved by Alice in 5 years is $6580.83.
Explanation of Solution
Given:
From part a the formula for
Where,
Calculation:
The total number of months in 5 years is 60.
Substitute 60 for n in equation (3), to find the value of total saved amount after 5 years.
Simplify the above expression,
Therefore, the total amount saved by Alice in 5 years is $6580.83.
Chapter 12 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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