Sequences and Series is the gateway topic of IB Mathematics Analysis & Approaches HL: it appears on every Paper 1 and almost every Paper 2, and it underpins later HL work on the binomial series, financial mathematics and Maclaurin expansions. Most lost marks come not from forgetting formulas (the AP and GP formulas are in the booklet) but from three sticky details — mis-stating the validity condition $|r| < 1$ for $S_\infty$, flipping the inequality wrongly when taking logs of a fraction less than one, and confusing real value with the TVM Solver in inflation problems.
This cheatsheet condenses every formula, identity, trick and trap from SL 1.2, 1.3, 1.4 and 1.8 into a single revision page. Scroll to the bottom for the printable PDF, the full Notes, the worked tutorials and the marked-up solutions in the gated student library.
§1 — Arithmetic Sequence (AP) SL 1.2
Core formulas
$n$th term:$u_n = u_1 + (n-1)d$ (booklet)
Sum of $n$:$S_n = \dfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr)$ (booklet)
Alt. sum:$S_n = \dfrac{n}{2}(u_1 + u_n)$ (if $u_n$ known)
Common difference:$d = u_{n+1} - u_n$ (constant)
$d$ from two terms:$d = \dfrac{u_n - u_m}{n - m}$ (not in booklet)
Key identity:$u_n = S_n - S_{n-1}$, $u_1 = S_1$
TrapIt is $(n-1)d$, NOT $nd$. The $u_1$ counts as the "zeroth jump". $n \in \mathbb{Z}^+$, so for "first $n$ such that $u_n > k$" you must round up.
Trap$u_n = S_n$ only when $n = 1$. For all other $n$, you need $u_n = S_n - S_{n-1}$.
TrickMaximum $S_n$ when $d < 0$: keep adding terms while $u_n \geq 0$. The largest $S_n$ occurs at the last non-negative term.
TrickThree-term AP test: $u_2 - u_1 = u_3 - u_2$, equivalently $2u_2 = u_1 + u_3$.
§2 — Geometric Sequence (GP) SL 1.3
Core formulas
$n$th term:$u_n = u_1 \cdot r^{n-1}$ (booklet)
Sum ($r \neq 1$):$S_n = \dfrac{u_1(r^n - 1)}{r - 1} = \dfrac{u_1(1 - r^n)}{1 - r}$ (booklet)
Common ratio:$r = \dfrac{u_{n+1}}{u_n}$ (constant)
$r$ from two terms:$r = \left(\dfrac{u_n}{u_m}\right)^{\frac{1}{n-m}}$
Three-term GP test:$a, b, c \text{ in GP} \iff b^2 = ac$
Edge case:If $r = 1$ then $S_n = n \, u_1$.
TrapThe exponent is $r^{\mathbf{n-1}}$, not $r^n$. Same off-by-one as the AP formula.
TrapNegative $r$ makes terms alternate in sign. Don't drop the alternation when using sums.
Note$r$ can be irrational (e.g. $r = \sqrt{3}$). Don't assume $r$ must be a "nice" rational.
§3 — Infinite GP & Convergence SL 1.8
Sum to infinity:$S_\infty = \dfrac{u_1}{1 - r}$, valid for $|r| < 1$ (booklet)
Tail formula:$S_\infty - S_n = \dfrac{u_1\, r^n}{1 - r}$ (not in booklet)
Use the tail for threshold problems: $\dfrac{u_1 r^n}{1 - r} < \varepsilon$ then take logs to solve for $n$.
NoteLog inequality rule: if $0 < r < 1$ then $\ln r < 0$, so $r^n < C \;\Rightarrow\; n > \dfrac{\ln C}{\ln r}$ — the inequality flips when you divide by a negative.
TrapAlways state "$|r| < 1$" before applying $S_\infty$. The IB awards a dedicated mark for this; omitting it is the most common dropped mark in this section.
TrickRecurring decimals are infinite GPs. $0.\overline{47} = \dfrac{47/100}{1 - 1/100} = \dfrac{47}{99}$.
§4 — Compound Interest & Depreciation SL 1.4
Compound:$FV = PV\!\left(1 + \dfrac{r}{100k}\right)^{\!kn}$ (booklet)
Depreciation:$FV = PV\!\left(1 - \dfrac{r}{100}\right)^{\!n}$ (use $I\% = -r$ in TVM)
| Symbol | Meaning |
|---|---|
| $FV$ | future value |
| $PV$ | present value |
| $r$ | annual rate (%) |
| $k$ | compounding periods per year |
| $n$ | years |
| Compounding frequency | $k$ |
|---|---|
| Annual | 1 |
| Half-yearly | 2 |
| Quarterly | 4 |
| Monthly | 12 |
NoteCompound interest is just a GP with ratio $\left(1 + \dfrac{r}{100k}\right)$. Recognising this is a fast way to attack disguised finance questions.
Real value (inflation-adjusted)
Real value:$\text{Real} = PV\!\left(\dfrac{1 + i/100}{1 + j/100}\right)^{\!n}$
$i$ is the investment rate per period, $j$ is the inflation rate per period.
TrapDo not use the Finance Solver for real-value problems — it cannot model the inflation correction. Use the formula above and compute by hand or on the home screen.
Trap$N = k \times \text{years}$ in the Finance Solver — not just years. For a 5-year monthly loan, $N = 60$, not $5$.
TrapEnter $r$ as a percentage (e.g. $5$ for 5%), not as a decimal $0.05$.
TrapRound $N$ up for "first time the value exceeds…"; round normally for everything else.
§5 — TI-Nspire TVM Solver Paper 2
Access: Menu → 8: Finance → 1: Finance Solver (Paper 2 only — not allowed on Paper 1).
| Field | Meaning |
|---|---|
| $N$ | total periods (= $k \times $ years) |
| $I\%$ | annual rate (%) |
| $PV$ | present value |
| $Pmt$ | payment per period |
| $FV$ | future value |
| $PpY$ | payments per year |
| $CpY$ | compounds per year |
Sign convention: money OUT of your pocket → negative (a deposit you make). Money IN → positive (interest you receive). $PV$ and $FV$ must always have opposite signs.
How to use: Enter all known values, leave the unknown blank. Cursor onto the unknown field and press Enter — the calculator solves it.
§6 — AP vs GP — Identification & "Find $k$" SL 1.2–1.3
Identification tests
- AP test: $u_2 - u_1 = u_3 - u_2$, equivalently $2u_2 = u_1 + u_3$.
- GP test: $\dfrac{u_2}{u_1} = \dfrac{u_3}{u_2} \iff u_2^2 = u_1 u_3$.
"Find $k$" 4-step method
- Write the AP or GP condition explicitly.
- Expand and simplify.
- Solve (often a quadratic in $k$).
- Check validity of each root — reject any that violate the problem's constraints (e.g. $r > 0$, $u_n > 0$).
§7 — Sums from $S_n$ Formula SL 1.2
If a question gives you $S_n$ as a closed-form expression and asks you to identify the sequence:
$n$th term:$u_n = S_n - S_{n-1}$ (for $n \geq 2$)
First term:$u_1 = S_1$
Check AP:$d = u_2 - u_1$
Common $S_n$ types in IB
- $S_n = an^2 + bn$ → AP with $d = 2a$ and $u_1 = a + b$.
- $S_n = \displaystyle\sum a r^k$ → GP, read off $r$ directly.
§8 — Exam Attack Plan All sections
| If you see… | Reach for… |
|---|---|
| "$S_n$ formula given" | $u_n = S_n - S_{n-1}$ |
| "Sum to infinity" | $\dfrac{u_1}{1-r}$, state $|r| < 1$ |
| "$S_\infty - S_n < \varepsilon$" | Tail formula + logs (flip!) |
| Recurring decimal | Infinite GP |
| "Compound interest" | Formula or TVM (P2) |
| "Real value" + inflation | Real multiplier (NO solver) |
| "AP and GP" mixed | Equate both conditions |
| "Show 3 terms in GP" | Show $u_2^2 = u_1 u_3$ |
| "$k$th term $= 0$" | Solve $u_1 + (k-1)d = 0$ |
| "Smallest $n$ such that…" | Solve inequality, $n \in \mathbb{Z}^+$ |
| "Depreciation" | $FV = PV(1 - r/100)^n$ |
| "Increases by %" | GP with $r = 1 + \%/100$ |
§9 — Formula Booklet Checklist Paper 1 & 2
In the booklet (just apply)
- $u_n = u_1 + (n-1)d$
- $S_n = \tfrac{n}{2}(2u_1 + (n-1)d)$
- $u_n = u_1 r^{n-1}$
- $S_n = \dfrac{u_1(r^n - 1)}{r - 1}$
- $S_\infty = \dfrac{u_1}{1 - r}$, $|r| < 1$
- $FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}$
NOT in the booklet (memorise)
- $u_n = S_n - S_{n-1}$
- $d = \dfrac{u_n - u_m}{n - m}$
- Tail: $S_\infty - S_n = \dfrac{u_1 r^n}{1 - r}$
- Real-value formula
- $b^2 = ac$ for GP test
- $\displaystyle\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2}$
Worked Example — IB-Style Infinite GP with Threshold
Question (HL Paper 1 style — 7 marks)
The first three terms of a geometric sequence are $24$, $-18$, and $13.5$.
- Find the common ratio $r$ and state why $S_\infty$ exists.
- Find $S_\infty$.
- Find the smallest value of $n$ such that $|S_\infty - S_n| < 0.01$.
Solution
- $r = \dfrac{-18}{24} = -\dfrac{3}{4}$. Since $|r| = \dfrac{3}{4} < 1$, $S_\infty$ exists. (M1)(A1)(R1)
- $S_\infty = \dfrac{u_1}{1 - r} = \dfrac{24}{1 - (-3/4)} = \dfrac{24}{7/4} = \dfrac{96}{7} \approx 13.71$. (A1)
- Tail: $|S_\infty - S_n| = \left|\dfrac{u_1 r^n}{1 - r}\right| = \dfrac{24 \cdot (3/4)^n}{7/4} = \dfrac{96}{7}(3/4)^n$. (M1)
- Set $\dfrac{96}{7}(3/4)^n < 0.01 \;\Rightarrow\; (3/4)^n < \dfrac{0.07}{96} \approx 7.292 \times 10^{-4}$. (M1)
- Take $\ln$: $n \ln(3/4) < \ln(7.292\times 10^{-4})$. Since $\ln(3/4) < 0$, the inequality flips: $n > \dfrac{\ln(7.292\times 10^{-4})}{\ln(3/4)} \approx 25.07$. So $n = 26$. (A1)
Examiner's note: The most common errors here are (i) forgetting the absolute value bars when $r$ is negative, and (ii) failing to flip the inequality when dividing by $\ln(3/4)$, which is negative. Both errors are catastrophic and lose all subsequent A marks.
Common Student Questions
How do I tell quickly whether a sequence is AP or GP?
Test the differences first: if $u_2 - u_1 = u_3 - u_2$, it is an AP. If not, test the ratios: $u_2 / u_1 = u_3 / u_2$ (equivalently $u_2^2 = u_1 \cdot u_3$) means it is a GP. If neither holds, it is neither — and if both hold, every term is equal (a constant sequence). Always do the AP test first because it is one subtraction; the GP test needs a division and a square.
When is the infinite sum $S_\infty$ valid, and what do I have to write?
$S_\infty = \dfrac{u_1}{1 - r}$ is only valid when $|r| < 1$. You must state this condition explicitly in your working — examiners award a dedicated mark for it. If $|r| \geq 1$ the series diverges and $S_\infty$ does not exist. Recurring decimals like $0.\overline{7}$ are perfect examples of valid infinite GPs (here $r = 0.1$).
How do I solve "find the smallest $n$ such that $S_\infty - S_n < \varepsilon$"?
Use the tail formula $S_\infty - S_n = \dfrac{u_1 \cdot r^n}{1 - r}$. Set this less than $\varepsilon$ and take logs. Crucial: when $0 < r < 1$, $\ln r$ is negative — dividing by $\ln r$ flips the inequality. Then round $n$ up to the next integer because $n$ must be a positive integer and the inequality must remain satisfied.
What's the difference between Compound Interest and Real Value with inflation?
Compound interest uses $FV = PV(1 + r/100k)^{kn}$ where $k$ is compounding periods per year. Real value adjusts for inflation: $\text{Real} = PV \cdot \left(\dfrac{1 + i/100}{1 + j/100}\right)^n$, where $i$ is the investment rate and $j$ is the inflation rate. Do not use the TI-Nspire Finance Solver for real-value problems — it cannot handle the inflation correction.
What sign convention does the TVM Solver use on the TI-Nspire?
Money OUT of your pocket is negative (a deposit you make), money IN is positive (interest you receive). $PV$ and $FV$ must have opposite signs — if you deposit $PV = -2000$, the future value $FV$ will come back positive. Also: $N$ is total periods ($k \times$ years), and $I\%$ is the annual rate as a percentage, not a decimal.
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