IB AA: Number and Algebra

This topic establishes the algebraic foundations for the course. It begins with a review of exponent laws and extends them to logarithms, focusing on their properties and their use in solving equations. The main part of the chapter introduces sequences and series. Students will learn to identify arithmetic and geometric progressions, find any term in a sequence, and calculate the sum of a given number of terms.

A significant portion is dedicated to the applications of these concepts, particularly using geometric series to model financial situations like compound interest and annuities. The chapter also covers the sum to infinity for convergent geometric series. The final major sub-topic is the binomial theorem, which provides a method for expanding expressions of the form (a+b)^n. This involves working with combinations, often written as nCr.

The most challenging parts for students are typically the word problems involving sequences and series, which require careful translation of the scenario into mathematical terms. Manipulating logarithms, especially using the change of base rule, can also be difficult at first. A strong command of basic algebra, including index laws and solving equations, is the most important prerequisite for this chapter.

Formulas for IB AA: Number and Algebra

nth term of an arithmetic sequence: $u_n = u_1 + (n-1)d$ u_1 is the first term, d is the common difference.
Sum of n terms of an arithmetic series: $S_n = \frac{n}{2}(2u_1 + (n-1)d)$ This can also be written as S_n = \frac{n}{2}(u_1 + u_n).
nth term of a geometric sequence: $u_n = u_1 r^{n-1}$ u_1 is the first term, r is the common ratio.
Sum of n terms of a geometric series: $S_n = \frac{u_1(r^n - 1)}{r-1}$ This formula is for r ≠ 1.
Sum to infinity of a geometric series: $S_\infty = \frac{u_1}{1-r}$ This formula is valid only when |r| < 1.
Compound interest: $FV = PV \left(1 + \frac{r}{100k}\right)^{kn}$ PV is present value, r is annual rate, k is compounding periods per year, n is years.
Logarithm from exponentiation: $a^x = b \Leftrightarrow x = \log_a b$ Defines the relationship between exponents and logarithms.
Logarithm change of base: $\log_b a = \frac{\log_c a}{\log_c b}$ Allows conversion between logarithms of different bases.
Binomial coefficient: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ Also written as nCr, it is the number of ways to choose r items from n.
The binomial theorem: $(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$ For expansion of (a+b)^n where n is a positive integer.