Infinite geometric sequences and series

The first term is $a$. For consecutive non-zero terms, calculate the common ratio using later term divided by earlier term: $r=\frac{u_{n+1}}{u_n}$. Check more than one pair when deciding whether a displayed series is geometric.

A positive ratio keeps the signs the same. A negative ratio makes signs alternate. The magnitude $|r|$ tells us whether the term sizes shrink, stay constant, or grow.

The general term is $u_n=ar^{n-1}$. A necessary condition for any infinite series to converge is $u_n\to0$; for a non-zero geometric series, this happens exactly when $|r|<1$.

A non-zero infinite geometric series converges exactly when $|r|<1$, equivalently $-1<r<1$. Repeated multiplication then makes $r^n\to0$, so the terms shrink toward zero and the partial sums settle at a finite limit.

If $|r|>1$, term magnitudes grow. If $r=1$, the same non-zero term is added forever. If $r=-1$, the partial sums oscillate between two values. In every case $|r|\ge1$, a non-zero geometric series diverges.

A negative ratio does not automatically cause divergence. If $-1<r<0$, term signs alternate but their magnitudes shrink, so the partial sums oscillate ever closer to one limit.

The zero series $0+0+0+\cdots$ is a trivial exception that sums to zero. IB questions normally use a non-zero first term, for which $|r|<1$ is the complete convergence test.

For $r\ne1$, the finite geometric sum is $S_n=\frac{a(1-r^n)}{1-r}$. If $|r|<1$, then $r^n\to0$, so the limit is $S_\infty=\frac{a}{1-r}$.

The finite formula can be derived algebraically. Write $S_n=a+ar+\cdots+ar^{n-1}$, multiply by $r$, and subtract the aligned equations. All middle terms cancel, leaving $(1-r)S_n=a(1-r^n)$.

A positive ratio gives partial sums that move toward the limit from one side when $a>0$. A negative ratio often makes the partial sums alternate above and below the limit.

For $|r|<1$, the equation $S_\infty=\frac{a}{(1-r)}$ gives $a=S_\infty(1-r)$. If $S_\infty\ne0$, it also gives $r=1-\frac{a}{S_\infty}$. Check that a calculated ratio satisfies $|r|<1$. The zero series does not determine a unique ratio from its first term and sum.

A known term adds the equation $u_n=ar^{n-1}$. Use it together with the sum equation, then check all algebraic candidates because polynomial equations can produce more than one ratio.

When the displayed terms contain an unknown, form the ratio between consecutive terms. A true geometric series must use the same ratio throughout.

A convergent series has a finite sum and can be approximated by a large partial sum. The difference between the exact limiting sum and $S_n$ is the remainder $R_n$.

For a convergent geometric series, the first omitted term after $S_n$ is $ar^n$. The remaining tail is itself a geometric series, so $R_n=\frac{ar^n}{(1-r)}$.

A divergent series has no finite $S_\infty$. Writing a numerical value from $\frac{a}{(1-r)}$ when $|r|\ge1$ is an invalid application of the formula, not a strange sum.

Infinite geometric series model repeated contributions that become smaller by a constant factor. The model is appropriate only when the repetition can continue conceptually and the ratio has magnitude less than 1.

For a bouncing ball, count the initial drop once. Each rebound height is then travelled twice—upward and downward—so the distance model is $D=h+2hq+2hq^2+\cdots$.

Recurring decimals are infinite geometric series because each repeated digit block shifts by a fixed power of 10. Repeated-dose models work similarly when a constant fraction remains before each new dose.

Many applications repeatedly retain a fixed fraction of an existing amount and then add a fixed contribution. If $A_{n+1}=rA_n+d$ with $|r|<1$, the long-term level $L$ satisfies $L=rL+d$, so $L=\frac{d}{(1-r)}$.

The same result can be viewed as an infinite geometric series: a new contribution is followed by retained amounts with factors of $r$. Be clear about the observation time—immediately before or immediately after each new contribution—because the first term changes.

Perpetuities provide a financial application. If a payment $P$ is made at the end of every period and the effective interest rate per period is $i>0$, its present value is $\frac{P}{(1+i)}+\frac{P}{(1+i)^2}+\cdots=\frac{P}{i}$.

A valid model needs a constant ratio, an unlimited conceptual repetition, and a quantity whose contributions can be added. Check whether physical limits, changing rates, or a finite time horizon make the infinite model unrealistic.

A GDC has no special infinity term to enter. Use the exact formula after checking $|r|<1$, or calculate partial sums for increasing values of $n$ to observe the limiting behaviour.

A finite table of $S_n=\frac{a(1-r^n)}{(1-r)}$ illustrates numerical behaviour but does not prove convergence. Prove $r^n\to0$ when $|r|<1$. For $a\ne0$ and $-1<r<0$, compare consecutive odd and even indices, such as $n=9,10,11,12$, to see partial sums alternate around the limit.

When an unknown ratio produces a nonlinear equation, use a numerical solver or graph intersection. Restrict the search to $-1<r<1$, record the equation entered, and verify each reported root in the original conditions.

Keep exact fractions through the algebra. If the context requires a decimal, use full GDC precision and round only the final answer.

Attempt each problem before opening its worked solution. The set progresses from identifying convergence to solving unknowns, estimating remainders, and building applied models.

In every sum-to-infinity question, make the convergence condition visible. Exact values are preferred unless the question requests a decimal or contextual rounding.

Identify the first term and common ratio, decide whether the question concerns a term, finite partial sum, infinite sum, or remainder, and check convergence before using an infinite-sum formula.

For modelling, define what every term represents, state when it is measured, verify the constant-ratio assumption, and finish with units, accuracy, and a contextual interpretation.