IB Mathematics Applications & Interpretation Topic SL…

Why Geometric Sequences Matter

Some quantities do not grow by a fixed amount each step, but by a fixed factor: a population rising by 3% a year, an investment earning interest, or a disease spreading to a set multiple of new people each day.
A single common ratio captures that repeated multiplication, and lets us predict a far-off value or total a long run of terms in one calculation.

Note

These are exactly the patterns behind the financial models in SL 1.4, so the ideas here carry straight into compound interest and depreciation.

What Makes A Sequence Geometric

A sequence is an ordered list of numbers, and its entries are called terms, labelled $u_1, u_2, u_3, \dots$ by position.
Where an arithmetic sequence adds a fixed number each step, a geometric sequence multiplies by a fixed number each step.
That fixed multiplier is the common ratio $r$, and you find it by dividing any term by the one before it.
In symbols, $r = \dfrac{u_2}{u_1} = \dfrac{u_3}{u_2}$, and the ratio must be the same every time for the sequence to be geometric.

Example

Take $2, 6, 18, 54, \dots$; the ratios are $\tfrac{6}{2} = 3$, $\tfrac{18}{6} = 3$, $\tfrac{54}{18} = 3$.
The ratio is always 3, so it is geometric with $u_1 = 2$ and $r = 3$.
By contrast $2, 5, 9$ has ratios $2.5$ then $1.8$, which are not equal, so it is not geometric.

A Formula For The nth Term

Build the terms from the first one: $u_1$, then $u_1 r$, then $u_1 r^2$, then $u_1 r^3$.
The power of $r$ counts how many times you have multiplied, which is always one less than the position, so position $n$ carries $r^{n-1}$. $$ u_n = u_1 r^{\,n-1} $$
1. $u_n$ is the term you want,
2. $u_1$ the first term,
3. $r$ the common ratio,
4. $n$ the position.
The value of $r$ sets the behaviour: $r > 1$ gives growth, $0 < r < 1$ gives decay toward zero, and a negative $r$ makes the terms alternate in sign.

Example question

A geometric sequence has $u_1 = 2$ and $r = 3$. Find $u_6$, and find which term first exceeds 1000.

Solution

$u_6 = 2 \times 3^{5} = 2 \times 243 = 486$.
Testing terms, $u_7 = 1458 > 1000$ while $u_6 = 486 < 1000$, so the 7th term is the first to exceed 1000.

Summing A Geometric Series

A series is the sum of a sequence's terms, and $S_n$ is the sum of the first $n$ of them.
The derivation is a shift-and-subtract. Start by writing out the sum, then multiply the whole thing by $r$. $$ S_n = u_1 + u_1 r + \dots + u_1 r^{n-1}, \qquad rS_n = u_1 r + u_1 r^2 + \dots + u_1 r^{n} $$
Subtracting the second line from the first cancels every middle term, leaving only the ends. $$ S_n - rS_n = u_1 - u_1 r^{n} \;\Rightarrow\; S_n(1-r) = u_1(1 - r^{n}) $$ $$ S_n = \frac{u_1(1 - r^{n})}{1 - r} = \frac{u_1(r^{n} - 1)}{r - 1}, \quad r \neq 1 $$
We need $r \neq 1$ because the last step divides by $1 - r$, and dividing by zero is undefined; if $r = 1$ every term is equal, so the sum is just $n u_1$.

Example question

A geometric sequence has $u_1 = 2$ and $r = 3$. Find $S_6$.

Solution

Since $r > 1$, use $S_n = \dfrac{u_1(r^{n}-1)}{r-1}$ to keep the numbers positive.
$S_6 = \dfrac{2(3^{6}-1)}{3-1} = \dfrac{2(728)}{2} = 728$

Growth And Decay Models

Geometric sequences model anything that changes by a fixed percentage each period, such as population growth, the spread of a disease, or a value depreciating each year.
A growth of $p\%$ per period means $r = 1 + \tfrac{p}{100}$, and a decay of $p\%$ means $r = 1 - \tfrac{p}{100}$, which is the idea behind the financial models in SL 1.4.
Just like an arithmetic series, a geometric series can be written compactly with sigma notation as $\sum_{k=1}^{n} u_1 r^{k-1}$.

Example question

A town of $50\,000$ people grows by 3% each year.

(a) Write a model for the population $P$ after $n$ years.
(b) Estimate the population after 10 years, to the nearest hundred.

Solution

A 3% yearly rise means $r = 1.03$, taking the current population as the starting value.
(a) $P = 50\,000 \times 1.03^{n}$.
(b) $P = 50\,000 \times 1.03^{10} = 67\,195.8\dots \approx 67\,200$ people.

Exam technique

The formula booklet (SL 1.3) gives $u_n = u_1 r^{n-1}$ and both forms of $S_n$.
A very common task gives two terms and asks for $r$: divide one by the other so $u_1$ cancels.
Watch for a negative $r$, which can hide when a question only pins down a ratio of even-position terms.
Your GDC will evaluate powers and sums, but show $u_n = u_1 r^{n-1}$ or the $S_n$ formula with values substituted, because that set-up is the method mark.

Active recall

How do you test whether a sequence is geometric?
Why does the power of $r$ in $u_n$ equal $n-1$?
Walk through why $S_n(1-r) = u_1(1-r^n)$ after the shift-and-subtract.
What common ratio matches 5% annual decay?
Given a geometric sequence with $u_2 = 12$ and $u_4 = 108$, find $r$.

Why Geometric Sequences Matter

Some quantities do not grow by a fixed amount each step, but by a fixed factor: a population rising by 3% a year, an investment earning interest, or a disease spreading to a set multiple of new people each day.
A single common ratio captures that repeated multiplication, and lets us predict a far-off value or total a long run of terms in one calculation.

Note

These are exactly the patterns behind the financial models in SL 1.4, so the ideas here carry straight into compound interest and depreciation.

What Makes A Sequence Geometric

A sequence is an ordered list of numbers, and its entries are called terms, labelled $u_1, u_2, u_3, \dots$ by position.
Where an arithmetic sequence adds a fixed number each step, a geometric sequence multiplies by a fixed number each step.
That fixed multiplier is the common ratio $r$, and you find it by dividing any term by the one before it.
In symbols, $r = \dfrac{u_2}{u_1} = \dfrac{u_3}{u_2}$, and the ratio must be the same every time for the sequence to be geometric.

Example

Take $2, 6, 18, 54, \dots$; the ratios are $\tfrac{6}{2} = 3$, $\tfrac{18}{6} = 3$, $\tfrac{54}{18} = 3$.
The ratio is always 3, so it is geometric with $u_1 = 2$ and $r = 3$.
By contrast $2, 5, 9$ has ratios $2.5$ then $1.8$, which are not equal, so it is not geometric.

A Formula For The nth Term

Build the terms from the first one: $u_1$, then $u_1 r$, then $u_1 r^2$, then $u_1 r^3$.
The power of $r$ counts how many times you have multiplied, which is always one less than the position, so position $n$ carries $r^{n-1}$. $$ u_n = u_1 r^{\,n-1} $$
1. $u_n$ is the term you want,
2. $u_1$ the first term,
3. $r$ the common ratio,
4. $n$ the position.
The value of $r$ sets the behaviour: $r > 1$ gives growth, $0 < r < 1$ gives decay toward zero, and a negative $r$ makes the terms alternate in sign.

Example question

A geometric sequence has $u_1 = 2$ and $r = 3$. Find $u_6$, and find which term first exceeds 1000.

Solution

$u_6 = 2 \times 3^{5} = 2 \times 243 = 486$.
Testing terms, $u_7 = 1458 > 1000$ while $u_6 = 486 < 1000$, so the 7th term is the first to exceed 1000.

Summing A Geometric Series

A series is the sum of a sequence's terms, and $S_n$ is the sum of the first $n$ of them.
The derivation is a shift-and-subtract. Start by writing out the sum, then multiply the whole thing by $r$. $$ S_n = u_1 + u_1 r + \dots + u_1 r^{n-1}, \qquad rS_n = u_1 r + u_1 r^2 + \dots + u_1 r^{n} $$
Subtracting the second line from the first cancels every middle term, leaving only the ends. $$ S_n - rS_n = u_1 - u_1 r^{n} \;\Rightarrow\; S_n(1-r) = u_1(1 - r^{n}) $$ $$ S_n = \frac{u_1(1 - r^{n})}{1 - r} = \frac{u_1(r^{n} - 1)}{r - 1}, \quad r \neq 1 $$
We need $r \neq 1$ because the last step divides by $1 - r$, and dividing by zero is undefined; if $r = 1$ every term is equal, so the sum is just $n u_1$.

Example question

A geometric sequence has $u_1 = 2$ and $r = 3$. Find $S_6$.

Solution

Since $r > 1$, use $S_n = \dfrac{u_1(r^{n}-1)}{r-1}$ to keep the numbers positive.
$S_6 = \dfrac{2(3^{6}-1)}{3-1} = \dfrac{2(728)}{2} = 728$

Growth And Decay Models

Geometric sequences model anything that changes by a fixed percentage each period, such as population growth, the spread of a disease, or a value depreciating each year.
A growth of $p\%$ per period means $r = 1 + \tfrac{p}{100}$, and a decay of $p\%$ means $r = 1 - \tfrac{p}{100}$, which is the idea behind the financial models in SL 1.4.
Just like an arithmetic series, a geometric series can be written compactly with sigma notation as $\sum_{k=1}^{n} u_1 r^{k-1}$.

Example question

A town of $50\,000$ people grows by 3% each year.

(a) Write a model for the population $P$ after $n$ years.
(b) Estimate the population after 10 years, to the nearest hundred.

Solution

A 3% yearly rise means $r = 1.03$, taking the current population as the starting value.
(a) $P = 50\,000 \times 1.03^{n}$.
(b) $P = 50\,000 \times 1.03^{10} = 67\,195.8\dots \approx 67\,200$ people.

Exam technique

The formula booklet (SL 1.3) gives $u_n = u_1 r^{n-1}$ and both forms of $S_n$.
A very common task gives two terms and asks for $r$: divide one by the other so $u_1$ cancels.
Watch for a negative $r$, which can hide when a question only pins down a ratio of even-position terms.
Your GDC will evaluate powers and sums, but show $u_n = u_1 r^{n-1}$ or the $S_n$ formula with values substituted, because that set-up is the method mark.

Active recall

How do you test whether a sequence is geometric?
Why does the power of $r$ in $u_n$ equal $n-1$?
Walk through why $S_n(1-r) = u_1(1-r^n)$ after the shift-and-subtract.
What common ratio matches 5% annual decay?
Given a geometric sequence with $u_2 = 12$ and $u_4 = 108$, find $r$.

IB Mathematics Applications & Interpretation SL 1.3—geometric Sequences Notes

IB Mathematics Applications & Interpretation SL 1.3—geometric Sequences Notes

Why Geometric Sequences Matter

What Makes A Sequence Geometric

A Formula For The nth Term

Summing A Geometric Series

Growth And Decay Models

Why Geometric Sequences Matter

What Makes A Sequence Geometric

A Formula For The nth Term

Summing A Geometric Series

Growth And Decay Models

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation SL 1.3—geometric Sequences Notes

IB Mathematics Applications & Interpretation SL 1.3—geometric Sequences Notes

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

Why Geometric Sequences Matter

What Makes A Sequence Geometric

A Formula For The nth Term

Summing A Geometric Series

Growth And Decay Models

Why Geometric Sequences Matter

What Makes A Sequence Geometric

A Formula For The nth Term

Summing A Geometric Series

Growth And Decay Models