Geometric sequences and series

The first term is the value at position 1. To test a sequence, divide each later term by the preceding term. If all defined quotients agree, the sequence is geometric and that value is $r$.

A ratio can be greater than 1, between 0 and 1, negative, 1 or 0. A negative ratio makes signs alternate. If a term is zero, quotient calculations may be undefined, so use the repeated-multiplier description or recurrence relation instead.

Do not confuse a percentage rate with its multiplier. Growth of $8\%$ gives $r=1.08$; decay of $8\%$ gives $r=0.92$.

A geometric sequence can be defined recursively by giving a starting term and a rule for moving forward: $u_1=a$ and $u_{n+1}=ru_n$.

Recurrence notation mirrors a spreadsheet: enter the first value once, then multiply the preceding cell by the fixed ratio and fill downward.

A recurrence is excellent for generating nearby terms, but the explicit nth-term formula is faster for a distant term.

Starting at $u_1=a$, reaching $u_n$ requires $n-1$ multiplications by $r$. Therefore the explicit formula is $u_n=ar^{n-1}$.

The exponent is $n-1$, not $n$. Check the formula at $n=1$: it must give $u_1=ar^0=a$.

Keep full calculator precision during the calculation and round only the final contextual value as requested.

For positive $a$: if $r>1$, terms grow; if $0<r<1$, terms decay toward zero; and if $r=1$, the sequence is constant. If $r=0$, every term after the first is zero.

If $r<0$, signs alternate. When $-1<r<0$, magnitudes shrink toward zero; when $r<-1$, magnitudes grow. Negative ratios are mathematically valid but often unsuitable for populations, prices or lengths because those quantities cannot alternate in sign.

A sequence graph consists of discrete points at integer values of $n$, not a continuous curve. A smooth exponential curve can show the trend, but values between term numbers are not automatically part of the sequence.

To find the position of a given term, solve $ar^{n-1}=T$. Exact powers can sometimes be recognized. Otherwise use a GDC numerical solver, graph intersection or logarithms if appropriate.

For positive $a$, $r$ and $T$ with $r\ne1$, logarithms give $n=1+\frac{\ln(\frac{T}{a})}{\ln r}$. A term number must be a positive integer.

For threshold questions such as “first exceeds,” solve the inequality, move to the next valid integer if necessary, then verify that the previous term fails and the chosen term succeeds.

If two terms are known, $u_p=ar^{p-1}$ and $u_q=ar^{q-1}$. Dividing eliminates $a$: $\frac{u_q}{u_p}=r^{q-p}$.

After finding $r$, substitute into either term equation to find $a$. If $q-p$ is even, a positive equation may allow both a positive and a negative ratio; use the data or context to decide whether both are valid.

Do not round a ratio early. Small rounding errors become large when the ratio is raised to a high power.

A geometric series is formed by adding geometric-sequence terms: $S_n=a+ar+ar^2+\cdots+ar^{n-1}$. The symbol $S_n$ means the sum of exactly the first $n$ terms.

A term question and a total question are different. “Amount in year 10” requires one term; “total over the first 10 years” requires a series sum.

Always count the number of included terms carefully. From index $k=m$ through $k=n$, inclusive, there are $n-m+1$ terms.

For $r\ne1$, a finite geometric sum can be written in two equivalent forms: $S_n=\frac{a(1-r^n)}{1-r}$ or $S_n=\frac{a(r^n-1)}{r-1}$. Choose one form and keep numerator and denominator signs consistent.

If $r=1$, every term equals $a$, so $S_n=na$. The usual fraction would divide by zero and must not be used.

The finite-sum formula works for positive, fractional and negative ratios. A finite series does not require a convergence condition.

Begin with $S_n=a+ar+ar^2+\cdots+ar^{n-1}$. Multiplying by $r$ shifts every power: $rS_n=ar+ar^2+\cdots+ar^n$.

Subtracting gives $S_n-rS_n=a-ar^n$, so $S_n(1-r)=a(1-r^n)$. Dividing by $1-r$ produces the formula when $r\ne1$.

This derivation is also a useful memory aid: multiplying by the ratio makes the middle terms cancel.

When $S_n$ is known, substitute into the finite-sum formula and solve the resulting exponential equation for $n$. Use exact powers, a GDC solver, graph intersection or logarithms.

Because $n$ counts terms, it must be a positive integer. A non-integer solution may mean the target total is never attained exactly; a threshold question instead requires the first integer that satisfies an inequality.

Verify by substituting the final integer into the original sum formula, not only the rearranged calculator equation.

The capital Greek letter $\Sigma$ represents addition. A finite geometric sum may be written $\sum\limits_{k=0}^{n-1}ar^k$. Starting at $k=0$ makes the first term $ar^0=a$.

The index is a counter. Expand the first two and final terms before choosing a formula; this prevents off-by-one errors when the lower limit is not zero.

For $\sum\limits_{k=m}^{n}cr^k$, the first term is $cr^m$, the ratio is $r$, and there are $n-m+1$ terms.

An infinite geometric series continues without a final term. This is a relevant extension for repeated processes such as bouncing or recurring payments. It has a finite sum only when $|r|<1$.

If $|r|<1$, then $r^n\to0$ as $n\to\infty$. Taking the limit of the finite-sum formula gives $S_\infty=\frac{a}{1-r}$.

If $|r|\ge1$, the terms do not approach zero, so the series diverges. The condition is about $|r|$, not merely $r<1$: for example, $r=-2$ does not converge.

Repeated percentage change is geometric. An increase of $p\%$ uses the multiplier $1+\frac{p}{100}$, while a decrease uses $1-\frac{p}{100}$. The percentage rate and the multiplier are not the same number.

If $A_0$ is the amount at time $t=0$, then after $t$ complete periods $A_t=A_0r^t$. This time-zero form differs from sequence notation $u_n=ar^{n-1}$; choose one convention and label it clearly.

Growth and decay models are common for populations, infections, website traffic, depreciation, medicine concentration and repeated efficiency losses. State the units and whether the model is realistic over the whole interval.

Many AI SL applications ask when a quantity first exceeds or falls below a threshold. The calculator may return a decimal time, but discrete annual, monthly or daily data require an appropriate whole-number period.

If $A_t=A_0r^t$ uses $t=0$ for the initial value, then the first sequence term corresponds to time zero. If a question labels that initial value as $u_1$, the same physical time is $n=1$.

A model can also have a capacity, non-negativity restriction or practical stopping rule. Check these conditions before summing or extrapolating.

Real measurements rarely have identical consecutive ratios. For positive data, an endpoint estimate over positions $p$ to $q$ is $r\approx\left(\frac{u_q}{u_p}\right)^{\frac{1}{(q-p)}}$. This spreads the total multiplicative change evenly across the intervals.

Compare predicted and observed values using residuals: $e_n=\text{observed}-\text{predicted}$. Small, patternless residuals support the model; a curved pattern or steadily changing ratios suggests the model is missing structure.

Interpolation stays within the observed interval and is generally safer than extrapolation. A sensible conclusion states the prediction, units, rounding and at least one model limitation.

Technology is expected in AI SL, but it does not replace mathematical setup. Before using a table or solver, identify the initial value, ratio, index convention and whether the question asks for a term or a sum.

For a sequence table, enter $u(n)=ar^{n-1}$. For a time-zero model, enter $A(t)=A_0r^t$. Use integer inputs when the context is discrete.

For an unknown term number, graph the model and target together or use a numerical solver. For imperfect data, use exponential regression, then interpret the regression multiplier and inspect whether extrapolation is reasonable.

A strong applied solution follows a repeatable path: define variables and time origin, decide whether change is multiplicative, identify or estimate the ratio, choose a term or sum model, calculate, validate and interpret.

Check multiplicative change by comparing ratios, not differences. If a percentage change repeats, convert it to a multiplier before writing $A_t=A_0r^t$.

Finish with a sentence in context. Include units, appropriate rounding, the required integer decision and any practical reason the model might eventually fail.

Most lost marks come from indexing, using the percentage instead of the multiplier, confusing a term with a sum, applying the infinite formula without checking convergence, or accepting a non-integer term number.

A quick check is to substitute $n=1$ into an nth-term formula. It should return $a$. For a finite sum, test a small case by direct addition.

Write enough method for another student to reproduce the result: formula, substitution, unrounded technology result, required rounding and contextual statement.

Attempt each question before opening the steps. Questions 1–4 establish the foundations, 5–9 combine formulas, and 10–12 develop application and modelling judgment.

In an exam, give exact forms where sensible and numerical answers to the requested accuracy.

Use this final checklist to select a method quickly and present a complete AI SL solution.

A geometric model is powerful because one ratio summarizes repeated proportional change, but every applied conclusion depends on sensible indexing and model assumptions.