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Chapter 10: Problem 49

Annuity A deposit of 100 dollars is made at the beginning of each month for 5 years in an account that pays \(10 \%\) interest, compounded monthly. Use a symbolic algebra utility to find the balance \(A\) in the account at the end of the 5 years. $$A=100\left(1+\frac{0.10}{12}\right)+\cdots+100\left(1+\frac{0.10}{12}\right)^{60}$$

Short Answer

Expert verified
The balance \(A\) in the account at the end of the 5 years would be obtained after substituting the provided values in the formula and performing the calculations. The final result will be dependent on the precision of mathematical calculations.

Step by step solution

01

Understand the given problem and formula

A $100 deposit is being made each month for 5 years. The interest rate on the account is 10% compounded monthly. The formula for the future value of an annuity is given, and we need to plug in these variables correctly in the formula \(A=100(1+\frac{0.10}{12})+...+100(1+\frac{0.10}{12})^{60}\).
02

Simplify the formula

We need to simplify using the sum of a geometric sequence which is given in this formula \(A=\frac{P[(1+r)^n-1]}{r}\), where \(P=100, r=\frac{0.10}{12}, n=60\). Satoshi is making the deposit at the beginning of each month and so this is an annuity due, the formula for which is \(A=\frac{P[(1+r)^n-1]}{r}*(1+r)\).
03

Perform the calculations

Substitute the given values into the simplified formula and perform the calculations. \(A=\frac{100[(1+\frac{0.10}{12})^{60}-1]}{\frac{0.10}{12}}*(1+\frac{0.10}{12})\).
04

Simplification and Result

Further simplifying the equation gives the final amount in the account at the end of 5 years. In most cases, an approximate decimal will need to be given as a result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compound Interest
Compound interest is a fundamental concept in finance that allows your money to grow at a faster rate than simple interest. It's interest on interest, which means the interest you earn each month is added to the principal amount, and together they continue to earn more interest.
Think of it as a snowball effect. Each month, not only is your original deposit earning interest, but the interest you earned each previous month is also earning interest. In mathematical terms, the compound interest for an annuity is calculated based off the formula for the future value of an annuity due.
The monthly interest rate in our example is calculated by dividing the annual rate by 12 (the number of months in a year). So, for a 10% annual rate, the monthly rate is \(\frac{0.10}{12}\). This concept is crucial to understanding how your money grows in a compounding scenario.
Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the context of annuities and compound interest, this concept helps in calculating the future value when deposits are made periodically.
The future value of annuity due involves a geometric sequence because each deposit grows exponentially as time progresses. Each month's deposit grows by a factor of \((1 + r)\), where \(r\) is the monthly interest rate. For example, the first month's deposit grows for the full period of compounding, hence raised to the power of 60 in our formula.
Understanding geometric sequences enables you to calculate the future value of an annuity efficiently, by summing up all these exponentially growing terms using the geometric series formula.
Future Value of Annuity
The future value of an annuity is the total value of a series of regular deposits at the end of a specified period, considering the compound interest. This is particularly useful for saving plans, like retirement or education savings.
The formula for calculating this when dealing with an annuity due is slightly adjusted from the ordinary annuity because deposits are made at the start of each period. The formula becomes more efficient by taking into account that each deposit effectively gains an extra period of growth. The full formula is:
\[ A = \frac{P[(1+r)^n-1]}{r} \times (1+r) \]
Inserting the given values from our problem, the equation allows us to find out how much the regular $100 deposits will amount to over 5 years at a 10% annual interest rate compounded monthly.
Annuity Due
An annuity due is a series of equal payments made at the beginning of each period, as opposed to the end (which would be an ordinary annuity). This small shift in timing makes a significant difference in the overall future value.
Because each payment has one extra period to grow compared to an ordinary annuity, the future value of an annuity due is typically higher. The formula for calculating involves multiplying the standard future value of an ordinary annuity by an additional factor of \((1 + r)\) to account for this extra compounding period.
For example, if you deposit $100 at the start of each month, each deposit earns interest for the full duration it's in the account, maximizing the effect of compound interest over 5 years in our scenario. It's this timing that differentiates annuity due from other types of annuities, leading to more robust savings growth in regular deposit scenarios.

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Most popular questions from this chapter

Sales The sales \(a_{n}\) (in billions of dollars) of Wal-Mart from 1996 through 2005 are shown below as ordered pairs of the form \(\left(n, a_{n}\right),\) where \(n\) is the year, with \(n=1\) corresponding to \(1996 .\) (Source: Wal-Mart Stores, Inc.) $$ \begin{array}{l}{(1,104.859),(2,117.958),(3,137.634),(4,165.013)} \\\ {(5,191.329),(6,217.799),(7,244.524),(8,256.329)} \\\ {(9,285.222),(10,312.427)}\end{array} $$ (a) Use the regression feature of a graphing utility to find a model of the form \(a_{n}=b n^{3}+c n^{2}+d n+f\) \(n=1,2,3, \ldots, 10,\) for the data. Graphically compare the points and the model. (b) Use the model to predict the sales in the year 2012 .

The repeating decimal is expressed as a geometric series. Find the sum of the geometric series and write the decimal as the ratio of two integers. $$ 0 . \overline{81}=0.81+0.0081+0.000081+\cdots $$

Cost For a family of four, the average costs per week to buy food from 2000 through 2006 are shown in the table, where \(a_{n}\) is the average cost in dollars and \(n\) is the year, with \(n=0\) corresponding to 2000 . (Source: U.S. Department of Agriculture) $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {0} & {1} & {2} & {3} & {4} & {5} & {6} \\ \hline a_{n} & {161.3} & {168.0} & {171.0} & {174.6} & {184.2} & {187.1} & {190.4} \\ \hline\end{array} $$ (a) Use the regression feature of a graphing utility to find a model of the form \(a_{n}=k n+b, \quad n=0,1,2,3,4,5,6\) for the data. Use a graphing utility to plot the points and graph the model. (b) Use the model to predict the cost in the year 2012 .

Write the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !}=1-\frac{1}{2}+\frac{1}{6}-\frac{1}{24}+\frac{1}{120}-\cdots $$

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty} 4\left(\frac{1}{4}\right)^{n}=4+1+\frac{1}{4}+\frac{1}{16}+\cdots $$
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