Number and Algebra — IB Math AI SL Study Guide

For: IB Mathematics Applications & Interpretation SL candidates.

Covers: IB AI SL Topic 1 — scientific notation, arithmetic and geometric sequences and series, exponentials and logarithms, financial mathematics (compound interest, depreciation, annuities), basic systems of equations.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the IB AI SL style for educational use. They are not reproductions of past IBO papers.

1. Why Number & Algebra in AI SL

Topic 1 makes up about 15-18% of AI SL — a lighter topic compared to Calculus or Statistics, but the only topic that explicitly handles financial mathematics (compound interest, loan repayments, savings plans). Memorise the GDC functionality (TVM solver) — most financial problems are solved via your calculator.

2. Scientific notation and approximation

Numbers in IB SL frequently use scientific notation: $a\times 10^n$ with $1\le a<10$ and $n\in \mathbb{Z}$.

Round to 3 sig fig unless the question specifies otherwise. For very small or very large numbers (atomic mass, planetary distances), scientific notation prevents zero-counting errors.

3. Arithmetic sequences and series

Common difference $d$. Position-to-term: $u_n=u_1+(n-1)d$.

Sum of first $n$ terms: $$S_n=\frac{n}{2}[2u_1+(n-1)d]=\frac{n}{2}(u_1+u_n)$$

Real-world: equal monthly savings, depreciation by fixed amount, evenly-spaced cash flows.

4. Geometric sequences and series

Common ratio $r$. Position-to-term: $u_n=u_1\cdot r^{n-1}$.

Sum of first $n$ terms (when $r\ne 1$): $$S_n=\frac{u_1(r^n-1)}{r-1}$$

Sum to infinity (when $|r|<1$): $S_{\infty }=\frac{u_1}{1-r}$.

Real-world: compound interest, exponential decay (radioactive, drug elimination), repeated bouncing ball.

5. Compound interest

$FV=PV\cdot (1+r/k{)}^{kn}$, where $PV$ is present value, $r$ is annual interest rate, $k$ is compoundings per year, $n$ is years.

Special cases:

• Annual: $FV=PV(1+r{)}^n$.
• Monthly: $FV=PV(1+r/12{)}^{12n}$.
• Continuous: $FV=PV\cdot e^{rn}$.

For loans/annuities, use GDC's TVM (Time Value of Money) solver: input $N$, $I\%$, $PV$, $PMT$, $FV$, leave the unknown blank. Conventions on AI SL: cash out (deposit, payment) is negative, cash in (withdrawal, loan received) is positive.

6. Exponentials and logarithms

Exponential growth: $y=a\cdot b^x$ where $b>1$. Decay: $0<b<1$.

Logarithm definition: ${\log }_{a}x=y⟺a^y=x$. Base 10 (just $\log$) and base $e$ (just $\ln$) are the most common.

Laws:

• $\log (xy)=\log x+\log y$
• $\log (x/y)=\log x-\log y$
• $\log (x^n)=n\log x$
• ${\log }_{a}x=\frac{\log x}{\log a}$ (change of base)

Solve exponential equations by taking $\log$ of both sides: $5^x=30⟹x=\log 30/\log 5\approx 2.113$.

7. Worked Example

A bank offers 4.5% annual interest, compounded monthly. You deposit $5000todayandadd$200 at the end of each month for 5 years.

(a) Find the future value of just the original $5000 after 5 years. (b) Use the geometric series sum to find the future value of the monthly contributions only. (c) Use a TVM solver (or sum) to find total balance.

Solution.

(a) $FV=5000\cdot (1+0.045/12{)}^{12\cdot 5}=5000\cdot 1.00375^{60}=5000\cdot 1.2517=6258.50$.

(b) Each $200depositgrowsfor$(60 - k)$months.Sum:$\sum_{k=1}^{60} 200 \cdot 1.00375^{60-k}$. This is geometric: first term $200\cdot 1.00375^{59}$, ratio $1/1.00375$, 60 terms. Equivalently: $$F{V}_{\text{deposits}}=200\cdot \frac{1.00375^{60}-1}{0.00375}=200\cdot \frac{0.2517}{0.00375}=200\cdot 67.13=13425$$

(c) Total: $6258.50+13425=19684$ (approx). TVM solver with $PMT=-200$, $PV=-5000$, $r=4.5\%$, $N=60$, compoundings = 12 will return $FV\approx 19684$.

8. Common Pitfalls

• Mismatched compounding period: monthly compounding requires $r/12$ in the exponent base AND $12n$ in the exponent. Don't mix.
• TVM sign convention: cash out is negative. Inconsistency here gives wrong answers.
• Geometric series condition: $S_{\infty }$ formula requires $|r|<1$. With $|r|\ge 1$ the series diverges.
• Log base: $\log$ on a calculator is base 10; $\ln$ is base $e$. Don't confuse with computer science default base 2.

9. Practice Questions

1. The 5th term of an arithmetic sequence is 23 and the 12th term is 51. Find the 1st term and the sum of the first 20 terms.
2. A car valued at $20,000 depreciates 15% per year. Find its value after 8 years.
3. Solve $2\cdot 3^x=5\cdot 2^x$ for $x$.

10. Quick Reference Cheatsheet

• AP: $u_n=u_1+(n-1)d$, $S_n=\frac{n}{2}(2u_1+(n-1)d)$.
• GP: $u_n=u_1{r}^{n-1}$, $S_n=u_1(r^n-1)/(r-1)$, $S_{\infty }=u_1/(1-r)$ if $|r|<1$.
• Compound: $FV=PV(1+r/k{)}^{kn}$.
• Continuous: $FV=PV\cdot e^{rn}$.
• Log laws: $\log (xy)=\log x+\log y$, $\log x^n=n\log x$.
• Change of base: ${\log }_{a}x=\log x/\log a$.

11. What's Next

Number & Algebra in AI SL feeds Topic 4 (Statistics) for distributions like geometric and Poisson, and Topic 5 (Calculus) for limits like $e=\lim (1+1/n{)}^n$. Use Ollie for any specific TVM or log problem: "Set up this loan repayment in TVM" or "Why is my exponential equation answer different from the textbook?"