Geometric sequences and series

From $u_1$ to $u_n$, the sequence is multiplied by $r$ exactly $n-1$ times. Therefore the explicit formula is $u_n=u_1r^{n-1}$. It finds a distant term without generating every previous term.

For a nonzero first term, If $|r|>1$, term magnitudes grow. If $0<|r|<1$, term magnitudes shrink toward zero. If $r<0$, odd and even powers have different signs, so always keep the ratio inside parentheses when evaluating powers. If the first term is zero, the recurrence gives only zero terms for every finite ratio.

A geometric nth-term rule is exponential in $n$, unlike the linear nth-term rule of an arithmetic sequence. This is why repeated percentage change can become much larger or smaller than constant additive change over many periods.

For $r\ne0$, rearrange $u_n=u_1r^{n-1}$ as $u_1=\frac{u_n}{r^{n-1}}$. If two terms are known and $u_p\ne0$, division gives $r^{q-p}=\frac{u_q}{u_p}$. When $r=0$, terms after the first are zero and do not determine the first term.

Be careful when the index gap $q-p$ is even. If $\frac{u_q}{u_p}>0$, both a positive and a negative real ratio may satisfy an even power. Extra information, such as another term or a statement that $r>0$, may be needed to choose one.

For a positive ratio $r\ne1$, logarithms can solve for a term number: $n=1+\frac{\ln(\frac{u_n}{u_1})}{\ln r}$, provided the ratio $\frac{u_n}{u_1}$ is positive. A term position must be a positive integer. For negative ratios, logarithms of $r$ are not real, so use sign and integer-power reasoning instead.

In threshold questions, solve the equality to locate the boundary, then test neighbouring integer positions in the original inequality. The direction of an inequality can change when dividing by a negative logarithm such as $\ln(0.8)$, so checking terms is essential.

A geometric series is the sum of terms from a geometric sequence: $S_n=u_1+u_1r+u_1r^2+\cdots+u_1r^{n-1}$. For $r\ne1$, its finite sum is $S_n=\frac{u_1(1-r^n)}{1-r}$.

The equivalent form $S_n=\frac{u_1(r^n-1)}{r-1}$ often feels more convenient when $r>1$ because the factors $r^n-1$ and $r-1$ are positive (the sign of $u_1$ still affects the sum). Both forms give exactly the same result when used consistently.

If $r=1$, every term equals $u_1$, so the special case is $S_n=nu_1$. The usual fraction would divide by zero and must not be used.

Finite geometric sums work for ratios larger than 1, between 0 and 1, and negative. With a negative ratio the terms alternate, so preserve signs rather than adding absolute values.

An infinite geometric series has no final term. For a nonzero first term it converges exactly when $|r|<1$. Then $r^n\to0$, giving $S_\infty=\frac{u_1}{(1-r)}$. If the first term is zero, every term and partial sum is zero for any finite ratio.

If $|r|\ge1$ and $u_1\ne0$, the terms do not approach zero, so the series diverges and has no finite sum. The condition that terms approach zero is necessary for any infinite series to converge; for a geometric series, $|r|<1$ is also sufficient.

A negative ratio can still converge. For example, $9-3+1-\frac13+\cdots$ has $r=-\frac13$. The partial sums alternate around their limiting value while moving closer to it.

The amount left after the first $n$ terms of a convergent geometric series is the remainder $R_n=S_\infty-S_n=\frac{u_1r^n}{1-r}$. It is also an infinite geometric tail whose first omitted term is $u_1r^n$.

Each new region is half the previous one. Infinitely many shrinking additions fill the whole without the total exceeding 1.

A finite geometric series can be written as $\sum\limits_{k=0}^{n-1}u_1r^k$. Starting at $k=0$ produces the first term $u_1r^0=u_1$, and ending at $n-1$ produces exactly $n$ terms.

The equivalent one-based form is $\sum\limits_{k=1}^{n}u_1r^{k-1}$. The index letter is a dummy variable, but the limits and exponent must agree. An incorrect shift may change the included terms, their count, or both; check the first and last terms and calculate the count from the limits.

An infinite geometric series may be written $\sum\limits_{k=0}^{\infty}u_1r^k$. This notation does not guarantee convergence; the condition $|r|<1$ must still be checked before assigning the sum $\frac{u_1}{(1-r)}$.

Geometric models describe repeated proportional change. An increase of $p\%$ uses growth factor $1+\frac{p}{100}$; a decrease of $p\%$ uses decay factor $1-\frac{p}{100}$. Apply the factor once per stated time interval.

Check the time index. If $P_0$ is the amount now, then after $t$ periods $P_t=P_0r^t$. If the initial amount is labelled $u_1$, then the same values follow $u_n=u_1r^{n-1}$. Confusing time 0 with term 1 creates an off-by-one power.

A model can produce one future amount, a finite accumulated total, or an infinite total. Decide whether the question asks for a term or a sum. Also judge whether extrapolating repeated percentage change indefinitely is realistic in context.

Bouncing-ball distance is a classic infinite-series model. The initial drop is counted once, while each rebound height is travelled upward and downward, so rebound distances are usually doubled.

Use a calculator table to generate terms, locate a threshold, or check alternating signs. Enter the nth-term rule with the correct starting index. For a finite series, use the summation template or the closed formula; calculate both ways occasionally to check your setup.

When the unknown is in an exponent, use logarithms, a numerical solver, a graph or a table. Keep full calculator precision during working and round only the final contextual result. Term numbers and counts must satisfy positive-integer constraints.

For positive growth or decay, the logarithmic boundary is efficient. For negative ratios, real logarithms cannot directly solve $(-a)^n=b$; instead reason about parity and test integer powers.

Before accepting output, estimate the direction and scale. A positive quantity in a decay model should not increase; a convergent infinite sum should be larger than the first term when all terms are positive; and a finite sum should reflect the signs of its terms.

Attempt each question before opening the worked steps. The set moves from recognizing common ratios to finite and infinite series, logarithmic thresholds and IB-style modelling.

After each solution, identify the key decision: ratio or difference, $n-1$ exponent, term or sum, finite or infinite, convergence condition, or integer threshold.