1.3 Geometric sequences and series

When a quantity is multiplied by the same factor at each step, it forms a geometric sequence. A colony of bacteria in which every cell divides into two each generation doubles in size again and again: $1, 2, 4, 8, 16, \ldots$ The same pattern, with different factors, describes a population growing by a fixed percentage each year, the value of an investment earning compound interest, and the shrinking heights of a bouncing ball. A geometric sequence is a discrete model of exponential change, and it grows (or shrinks) far faster than an arithmetic sequence.

Geometric sequences

In the sequence $3, 6, 12, 24, \ldots$ each term is twice the one before. In $64, 32, 16, 8, \ldots$ each term is half the one before. Sequences like these, built by repeated multiplication, are geometric sequences, and the fixed multiplier is the common ratio.

In recursive form, a geometric sequence satisfies $u_{n+1} = r\,u_n$. To test whether a list of numbers is geometric, divide each term by the term before it: the sequence is geometric only if every ratio is the same. Neither $u_1$ nor $r$ can be zero, because a zero term would make every later term zero and the ratios undefined.

How the common ratio shapes a sequence

The value of $r$ decides how a geometric sequence behaves. For a positive first term:

A negative ratio makes the terms switch between positive and negative. A ratio between $-1$ and $1$ makes the terms shrink in size, which is the case that leads to infinite sums in 1.8.

The nth term of a geometric sequence

Writing the terms in terms of $u_1$ and $r$ shows the pattern. Each step multiplies by $r$ once, so the second term has one factor of $r$, the third has two, and the $n$th has $n - 1$.

The structure matches the arithmetic formula $u_n = u_1 + (n-1)d$: repeated addition of $d$ becomes repeated multiplication by $r$, so $(n-1)d$ becomes $r^{n-1}$.

Finding the first term and the common ratio

Given two terms, write each with $u_n = u_1 r^{n-1}$. Dividing one equation by the other cancels $u_1$ and leaves a power of $r$. This is the geometric version of subtracting the equations for an arithmetic sequence.

For three consecutive terms $a, b, c$ of a geometric sequence, the ratios are equal, so $\frac{b}{a} = \frac{c}{b}$, which gives $b^2 = ac$. This is the multiplicative version of $2b = a + c$ for arithmetic sequences.

Finding the number of terms

When the value of a term is known but its position $n$ is not, the unknown is in the exponent: $u_1 r^{n-1} = \text{value}$. If the value is an exact power, $n$ can be found by inspection. Otherwise use a GDC table of $u_n$ against $n$, or logarithms (see 1.7), which turn $r^{n-1} = c$ into $n - 1 = \frac{\log c}{\log r}$.

Questions about when a sequence first passes a value lead to an inequality instead of an equation. The solution of the matching equation is usually not a whole number, so the answer is the next whole number above it, confirmed by checking the terms on either side.

Geometric series

The sum of the terms of a geometric sequence is a geometric series. As with arithmetic series, there is a formula for the sum of the first $n$ terms, but the trick that finds it is different: the series is multiplied by $r$ and subtracted from itself.

Deriving the sum formula

Write the sum, then multiply every term by $r$. Each term moves one place along, so the two lines share all but their first and last terms.

Subtracting the first line from the second cancels every shared term.

Dividing by $r - 1$, which is allowed only if $r \ne 1$, gives the sum formula. Multiplying the top and bottom by $-1$ gives an equivalent second form.

Both forms give the same value. The first is neater when $r > 1$ and the second when $r < 1$, since each then has a positive numerator and denominator. When $r = 1$ every term equals $u_1$, so $S_n = n u_1$.

Sigma notation for geometric series

Sigma notation, introduced for arithmetic series in 1.2, works in the same way for geometric series. A sum such as $\sum_{k=1}^{n} a r^{k-1}$ has first term $a$ (when $k = 1$), ratio $r$ and $n$ terms. When the expression is $a r^k$ instead, or the lower limit is not 1, write out the first two or three terms to find $u_1$ and $r$, and count the terms as upper limit minus lower limit plus one.

Finding n or r from a sum

When the sum is given, the sum formula becomes an equation for the unknown. If $n$ is unknown, it is in an exponent, so use a GDC table, a GDC solver or logarithms. If $r$ is unknown and $n$ is small, writing out the sum often gives a polynomial equation in $r$.

Arithmetic or geometric?

Data and patterns do not come labelled. To decide which model fits, calculate both the differences and the ratios of consecutive values. Constant differences show an arithmetic sequence; constant ratios show a geometric sequence. If neither is constant, the sequence is neither, although one of them may still be a good approximate model.

The two kinds of growth start in a similar way, but their long-term behaviour is very different. An arithmetic sequence with $d > 0$ adds the same amount every step. A geometric sequence with $r > 1$ adds more each step, because the increase is a fixed percentage of an ever larger value, so it eventually overtakes any arithmetic sequence.

Applications of geometric sequences

Whenever a quantity changes by the same percentage in each time period, its values form a geometric sequence. An increase of $p\%$ multiplies the quantity by $1 + \frac{p}{100}$, and a decrease of $p\%$ multiplies it by $1 - \frac{p}{100}$. So a growth of $3\%$ gives $r = 1.03$, and a loss of $25\%$ gives $r = 0.75$.

Counting periods correctly

In applications, it matters whether the starting value counts as $u_1$ or as "time zero". If a population starts at $P$ and grows by a factor $r$ each year, then after $1$ year it is $Pr$, after $2$ years $Pr^2$, and after $t$ years $Pr^t$. With $u_1 = P$ as the starting value, the value after $t$ years is the term $u_{t+1} = Pr^t$.

Population growth and cell division

Bacteria such as E. coli can divide about every 20 minutes under ideal laboratory conditions. Each division doubles the number of cells, so the counts form a geometric sequence with $r = 2$, and in a few hours a small sample becomes millions of cells.

The wheat and chessboard problem

A well-known puzzle places one grain of wheat on the first square of a chessboard, two on the second, four on the third, and so on, doubling on each of the 64 squares. The numbers of grains form a geometric sequence with $u_1 = 1$ and $r = 2$.

The result in part (b) holds for doubling in general: the sum of all the terms so far is one less than the next term, since $1 + 2 + 4 + \cdots + 2^{n-1} = 2^n - 1$. Doubling therefore produces more in its last step than in all the earlier steps combined.

The spread of a virus

In the early stages of an outbreak, when almost everyone is still susceptible, each infected person passes the virus on to roughly the same number of new people. The numbers of new cases in successive generations of infection then form an approximately geometric sequence.

Bouncing balls

A ball dropped onto a hard floor rises to a fixed fraction of its previous height after each bounce. The bounce heights form a geometric sequence with $0 < r < 1$, so they decrease towards zero.

The total vertical distance the ball travels, if it keeps bouncing, needs an infinite geometric series and is found in 1.8.

Exam-style questions on geometric sequences

Geometric sequences appear on both papers. Non-calculator questions use exact powers and fractions; calculator questions are usually set in context and need a GDC table, solver or logarithms to find $n$.

Self-check

• To do:

  I can decide whether a sequence is geometric and find its common ratio, including negative and fractional ratios.

• To do:

  I can explain why $u_n = u_1 r^{n-1}$ and use it to find any term.

• To do:

  I can find $u_1$ and $r$ from two terms, and give both values of $r$ when the power is even.

• To do:

  I can derive the sum formula by multiplying by $r$ and subtracting, and use both forms.

• To do:

  I can evaluate geometric sums in sigma notation, including sums that start at $k = 0$.

• To do:

  I can find the number of terms, or the first term or sum that passes a value, using a GDC table or logarithms.

• To do:

  I can tell arithmetic and geometric patterns apart from data, using differences and ratios.

• To do:

  I can model growth and decay with geometric sequences and comment on the limits of the model.

(b) Here $u_1 = 48$ and $r = -\frac{1}{2}$.

The odd power keeps the negative sign, which fits the pattern: even-numbered terms are negative.

If $x = 6$, the terms are $6, 12, 24$ and $r = 2$. If $x = -2$, the terms are $-2, 4, -8$ and $r = -2$.

(b) Here $u_1 = 64$, $r = \frac{1}{2}$ and $n = 8$. Use the second form since $r < 1$.

(c) This sequence has 8 terms (found in the earlier example), with $u_1 = 2$ and $r = 3$.

(b) When $k = 0$ the term is $2(1.5)^0 = 2$, so $u_1 = 2$ and $r = 1.5$. There are $9 - 0 + 1 = 10$ terms.

A GDC summation of $2(1.5)^k$ from $k = 0$ to $9$ confirms the value.

Since $3^6 = 729$ and $3^7 = 2187$, the smallest $n$ is $7$. Check: $S_6 = 728$ and $S_7 = 2186$. (b) Write the sum of three terms directly.

So $r = 3$ (giving $5 + 15 + 45$) or $r = -4$ (giving $5 - 20 + 80$). Both sums equal $65$.

This is an estimate: the ratios vary, and growth cannot continue at this rate indefinitely because nutrients and space run out.

Then $u_1 = \frac{-6}{-3} = 2$. (An odd power has only one real root, so there is one answer.) (b) Use the sum formula with $u_1 = 2$, $r = -3$, $n = 6$.

(c) Substitute into the second form of the formula and simplify.

Check with $n = 6$: $\frac{1 - 729}{2} = -364$, as in (b).