College Algebra Study Notes: Sequences, Series, and Financial Applications

Sequences, Series, and Financial Applications

Arithmetic Sequences and Series

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted by d.

• General Term (nth term): The nth term of an arithmetic sequence is given by:

$$ a_n = a_1 + (n-1)d$$

• Sum of the First n Terms: The sum of the first n terms of an arithmetic sequence is:

$$ S_n = \frac{n}{2}(a_1 + a_n)$$

• Example: If $a_1 = 3$ and $d = 2$, the 5th term is $a_5 = 3 + (5-1)\times2 = 11$.

Geometric Sequences and Series

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted by r.

• General Term (nth term):

$$ a_n = a_1 r^{n-1}$$

• Sum of the First n Terms:

$$ S_n = a_1 \frac{1 - r^n}{1 - r}, \quad r \neq 1$$

• Sum of an Infinite Geometric Series: If $|r| < 1$:

$$ S = \frac{a_1}{1 - r}$$

• Example: For $a_1 = 2$, $r = 3$, the 4th term is $a_4 = 2 \times 3^{3} = 54$.

Financial Applications: Present and Future Value

Sequences and series are used in finance to calculate the present and future value of investments and loans.

• Future Value of an Ordinary Annuity: The future value $F$ of an ordinary annuity (equal payments made at the end of each period) is:

$$ F = P \frac{(1 + r)^n - 1}{r}$$

• Present Value of an Ordinary Annuity: The present value $PV$ is:

$$ PV = P \frac{1 - (1 + r)^{-n}}{r}$$

• Example: If $P = 1000$, $r = 0.05$, $n = 10$, then $F = 1000 \frac{(1.05)^{10} - 1}{0.05}$.

Binomial Theorem and Binomial Coefficients

The Binomial Theorem provides a formula for expanding powers of binomials. The coefficients in the expansion are called binomial coefficients and are denoted $\binom{n}{k}$.

• Binomial Theorem:

$$ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

• Binomial Coefficient:

$$ \binom{n}{k} = \frac{n!}{k!(n-k)!}$$

• Example: The coefficient of $x^2$ in $(x + 2)^4$ is $\binom{4}{2} \times 1^2 \times 2^2 = 6 \times 1 \times 4 = 24$.

Application: Investment and Loan Problems

Many real-world problems involve finding how much to invest now (present value) or how much will be accumulated in the future (future value) using geometric series formulas.

• Example Problem: If an investment account earns 5% interest compounded annually, and you want $10,000$ in 10 years, how much should you invest now?

$$ PV = \frac{10000}{(1.05)^{10}}$$

• Example Problem: If you deposit $500$ each year into an account earning 6% interest, how much will you have after 8 years?

$$ F = 500 \frac{(1.06)^8 - 1}{0.06}$$

Tables: Binomial Coefficients and Sequences

The following table summarizes binomial coefficients for $n = 0$ to $n = 5$:

n	k = 0	k = 1	k = 2	k = 3	k = 4	k = 5
0	1
1	1	1
2	1	2	1
3	1	3	3	1
4	1	4	6	4	1
5	1	5	10	10	5	1

Additional info: The handwritten work on the page shows step-by-step calculations for arithmetic and geometric sequences, binomial coefficients, and financial mathematics problems, including present and future value calculations. These are all core topics in College Algebra, especially in the chapters on Sequences, Series, and Induction, and Financial Applications.