The annual profit for the H. J. Heinz Company from 1980 through 1989 can be approximated by the model $a_n=167.5 e^{0.12 n}$, n=0,1,2, ..., 9 where $a_n$ is the annual profit in millions of dollars and n represents the year, with n=0 corresponding to 1980 . Use the formula for the sum of a geometric sequence to approximate the total profit earned during this 10-year period.

A government program that currently costs taxpayers $2.5 billion per year is cut back by 20 percent per year. Write an expression for the amount budgeted for this program after n years.

You accept a job with a salary of $30,000 for the first year. Suppose, during the next 39 years, you receive a 5% raise each year. What would be your total compensation over the 40year period?

Use a Taylor polynomial to approximate the function with an error of less than 0.001.

ln (1.75)

A deposit of $100 is made at the beginning of each month in an account at an annual interest rate of 12% compounded monthly. The balance in the account after n months is

$$A_n=100(101)\left[(1.01)^n-1\right] .$$

Find the balance after 20 years by computing the 240th term of the sequence.

A deposit of $100 is made at the beginning of each month in an account at an annual interest rate of 12% compounded monthly. The balance in the account after n months is

$$A_n=100(101)\left[(1.01)^n-1\right] .$$

Find the balance after 5 years by computing the 60 th term of the sequence.

A deposit of $100 is made at the beginning of each month in an account at an annual interest rate of 12% compounded monthly. The balance in the account after n months is

$$A_n=100(101)\left[(1.01)^n-1\right] .$$

Compute the first six terms of the sequence $\left\{A_n\right\}$.

Use a Taylor polynomial to approximate the function with an error of less than 0.001.

cos (0.75)

Let $\Sigma a_n$ be a convergent series, and let

$$R_N=a_{N+1}+a_{N+2}+\cdots \cdot$$

be the remainder of the series after the first N terms. Prove that

$$\lim _{N \rightarrow \infty} R_N=0 .$$

Consider the sequence $\left\{A_n\right\}$ whose nth term is given by

$$A_n=P\left(1+\frac{r}{12}\right)^n$$

where P is the principal, $A_n$ is the amount at compound interest after n months, and r is the interest rate compounded annually. Find the first ten terms of the sequence if P=$9000 and r=0.115.

Consider the sequence $\left\{A_n\right\}$ whose nth term is given by

$$A_n=P\left(1+\frac{r}{12}\right)^n$$

where P is the principal, $A_n$ is the amount at compound interest after n months, and r is the interest rate compounded annually. Is $\left\{A_n\right\}$ a convergent sequence? Explain.

Use a Taylor polynomial to approximate the function with an error of less than 0.001.

sin 95$^{\circ}$

Show that the series

$$\sum_{n=1}^{\infty} a_n$$

can be written in the telescoping form

$$\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_n\right)\right]$$

where $S_0=0$ and $S_n$ is the nth partial sum.

Prove that every decimal with a repeating pattern of digits is a rational number.