Geometric Sequences
A video goes viral. In minute 1, one person shares it. In minute 2, 3 people see it. In minute 3, 9 people. Then 27… Every minute the number of new viewers is multiplied by 3. How many new viewers would that be in minute 20? Sequences that multiply by the same number each step are called geometric. They explode (or fade away) much faster than the "add the same amount" sequences you already know.
What is a geometric sequence?
Some textbooks call this a geometric progression. It is exactly the same thing.
The value of $r$ tells you how the sequence behaves:
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$r > 1$: it grows faster and faster ($3, 6, 12, 24, \dots$).
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$0 < r < 1$: it shrinks toward 0 ($64, 32, 16, 8, \dots$ with $r = \tfrac{1}{2}$).
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$r < 0$: the signs alternate ($1, -2, 4, -8, \dots$).
Slide the values and watch the pattern. Compare it with an arithmetic sequence, where the graph is a straight staircase. Here it takes off.
Analyze the following number sequences. Your task is to identify which of them are geometric progressions. For those that are, indicate their common ratio ($r$). a)…
Analyze the following number sequences. Your task is to identify which of them are geometric progressions. For those that are, indicate their common ratio ($r$).
- a) $2, 6, 18, 54, \dots$
- b) $5, 10, 15, 20, \dots$
- c) $80, 40, 20, 10, \dots$
- d) $3, -3, 3, -3, \dots$
Imagine that a viral video on a social network is shared very quickly. One person uploads the video (minute 0). In minute 1, 5 new people watch it. In minute 2, each of…
Imagine that a viral video on a social network is shared very quickly. One person uploads the video (minute 0). In minute 1, 5 new people watch it. In minute 2, each of those 5 people shares it with 5 other new people, and so on.
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Write the first four terms of the sequence of new viewers per minute.
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What is the common ratio $r$ of growth?
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Write the formula for the general term $a_n$.
We have a geometric progression made up of fractions. We know that the first terms are: $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots $$ a) Identify the…
We have a geometric progression made up of fractions. We know that the first terms are:
$$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots $$- a) Identify the first term $a_1$ and the common ratio $r$.
- b) Calculate the term $a_6$ without multiplying one by one, using the general formula.
The explicit formula
Look at how the terms are built from $a_1$:
$$a_2 = a_1 r, \qquad a_3 = a_1 r^2, \qquad a_4 = a_1 r^3, \quad \dots$$To reach $a_n$ you multiply by $r$ only $n - 1$ times (the first term needs no multiplication). That gives the explicit formula:
$$a_n = a_1 \cdot r^{\,n-1} \qquad\qquad \text{or, starting from any term: } a_n = a_k \cdot r^{\,n-k}$$Given a geometric progression where the first term is $a_1 = 3$ and the common ratio is $r = 2$: a) Calculate the value of the 7th term ($a_7$). b) Calculate the value…
Given a geometric progression where the first term is $a_1 = 3$ and the common ratio is $r = 2$:
- a) Calculate the value of the 7th term ($a_7$).
- b) Calculate the value of the 10th term ($a_{10}$).
Use the formula for the general term: $a_n = a_1 \cdot r^{n-1}$.
Look at the following figure. We start with a large square of side $8 \text{ cm}$. The next square is formed with a side length that is half of the previous one, and so…
Look at the following figure. We start with a large square of side $8 \text{ cm}$. The next square is formed with a side length that is half of the previous one, and so on.
Calculate the side length of the first 4 squares and write the general term $a_n$ of the sequence formed by these lengths.
A rubber ball is dropped from a certain height. On each bounce, the ball reaches a height that is half of the previous height. If on the first bounce ($a_1$) it reaches…
A rubber ball is dropped from a certain height. On each bounce, the ball reaches a height that is half of the previous height.
If on the first bounce ($a_1$) it reaches a height of $160 \text{ cm}$:
- a) Write the sequence of heights for the first 5 bounces.
- b) What is the common ratio $r$ of this progression?
- c) What height will it reach on the eighth bounce?
Finding the ratio from two terms
Often you do not get $a_1$ and $r$. You get two terms somewhere in the sequence. Use $a_n = a_k \cdot r^{\,n-k}$: the number of "jumps" between the two terms is the exponent of $r$. Then solve for $r$ with a root.
The same idea lets you insert numbers between two terms (these inserted numbers are called geometric means). If you insert $k$ numbers between $a_1$ and the last term, the sequence has $n = k + 2$ terms. Then use $a_n = a_1 r^{n-1}$ to find $r$.
A grandfather decides to save for his grandson following a curious mathematical rule. The first month he puts $\$2$ in a piggy bank. The second month he puts in $\$6$.…
A grandfather decides to save for his grandson following a curious mathematical rule. The first month he puts $\$2$ in a piggy bank. The second month he puts in $\$6$. The third month he puts in $\$18.$
If he continues with this logic:
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Show that it is a geometric progression and find the common ratio $r$.
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Calculate how much money he will have to put in the piggy bank in the fifth month.
In a role-playing video game, the experience needed to level up follows a geometric progression. We know that to reach level 2 you need 150 experience points, and to…
In a role-playing video game, the experience needed to level up follows a geometric progression. We know that to reach level 2 you need 150 experience points, and to reach level 5 you need 1200 points.
Calculate:
- a) The common ratio $r$ of the progression.
- b) The experience needed for the first level ($a_1$).
- c) The experience needed to reach level 10.
The sum of the first $n$ terms
Adding up the first $n$ terms of a sequence gives a series. For a geometric sequence there is a shortcut. Write $S_n = a_1 + a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1}$. Multiply by $r$ and subtract: almost every term cancels, leaving $S_n - rS_n = a_1 - a_1 r^n$. Solving for $S_n$ gives the formula.
Michael wants to buy an electric scooter that costs $\$400$. He decides to save following a strict plan: the first week he saves $\$2$, the second week $\$4…
Michael wants to buy an electric scooter that costs $\$400$. He decides to save following a strict plan: the first week he saves $\$2$, the second week $\$4$, the third $\$8$, and so on, doubling the amount each week.
- a) How much money will he have to save in week number 8?
- b) How much money will he have accumulated in total by the end of week 8?
- c) Will he have saved enough money to buy the scooter by the end of week 8? If not, how many weeks does he need in total?
We have a square with a side length of 10 inches. We join the midpoints of its sides to form a second inscribed square. We repeat the process with the second square to…
We have a square with a side length of 10 inches. We join the midpoints of its sides to form a second inscribed square. We repeat the process with the second square to form a third, and so on.
- a) Knowing that the area of each new square is half of the previous one, write the sequence of the areas of the first 5 squares.
- b) Is it an arithmetic or geometric progression? Identify the first term and the common difference or common ratio.
- c) Calculate the sum of the areas of the first 10 generated squares.
We want to interpolate (insert) 4 numbers between 3 and 96 to form a geometric progression of 6 terms in total. That is, we are looking to construct the sequence: $$ 3,…
We want to interpolate (insert) 4 numbers between 3 and 96 to form a geometric progression of 6 terms in total. That is, we are looking to construct the sequence:
$$ 3, \quad x_1, \quad x_2, \quad x_3, \quad x_4, \quad 96 $$-
Identify which position 3 and 96 occupy in the progression ($a_1$ and $a_n$).
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Use the formula for the general term to rearrange and calculate the common ratio $r$.
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Write the four numbers we have interpolated.
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If we wanted to sum all these 6 numbers, what would be the result?
When the rule uses a running total
Some problems do not tell you "multiply by $r$." They describe each day's amount using the total so far. Call that total $S_n$ (the sum of the first $n$ terms). The trick is to write the rule for day $n$ and for day $n-1$, then subtract: $S_n - S_{n-1} = a_n$ turns the running total back into single terms.
For the exercise below, translate each sentence into an equation with $S_n$, compute the amounts for the first days directly from the wording, and then check the ratios before you add them up.
A famous YouTuber* launches a charity campaign. On the first day he donates $\$10.…
A famous YouTuber launches a charity campaign. On the first day he donates $\$10.$ He commits that on each subsequent day he will donate an amount such that the total sum donated up to that day is triple what he donated individually the previous day.
Note: Read the question carefully, it is a mathematical riddle about the common ratio $r$.
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Calculate how much he donates on the second and third day.
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Show that the amounts donated day by day form a geometric progression and find its common ratio $r$.
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How much money will he have donated in total after a week (7 days)?
The product of terms
Some problems ask for the product of the first terms instead of the sum. Terms that are the same distance from the two ends have the same product. In $a_1, a_2, \dots, a_6$:
$$a_1 \cdot a_6 = a_2 \cdot a_5 = a_3 \cdot a_4$$Why? Each product is $a_1^2 \cdot r^5$, because the exponents always add up to 5. So the six terms form 3 equal pairs:
$$P_n = (a_1 \cdot a_n)^{n/2}$$To use this you often need $r$ first. Find it from the given terms as in the earlier section.
Calculate the product of the first 6 terms of a geometric progression knowing that the first term is $a_1 = 3$ and that the fourth term is $a_4 = 24$. Hint: First you…
Calculate the product of the first 6 terms of a geometric progression knowing that the first term is $a_1 = 3$ and that the fourth term is $a_4 = 24$.
Hint: First you will need to use the data to find the common ratio $r$ of the progression.
We have a geometric progression of which we do not know the common ratio $r$ and the first term $a_1$. However, we know that the third term is 12 ($a_3 = 12$) and the…
We have a geometric progression of which we do not know the common ratio $r$ and the first term $a_1$. However, we know that the third term is 12 ($a_3 = 12$) and the sixth term is 96 ($a_6 = 96$).
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Set up a system of equations or use the formula for the general term to find the common ratio $r$.
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Once you have the common ratio, calculate the first term $a_1$.
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Calculate the product of the first 6 terms of the sequence ($P_6$).
Infinite sums when $|r| < 1$
If the ratio is between $-1$ and $1$, the terms get closer and closer to 0. Surprisingly, the sum of all of them, even infinitely many, settles on a single number.
Picture a bar of length 1. Cover half of it, then half of what is left, then half of that, and so on. You never go past the end of the bar.
Check with the bar: $a_1 = \tfrac{1}{2}$ and $r = \tfrac{1}{2}$ give $S_\infty = \frac{1/2}{1 - 1/2} = 1$. ✓
Inscribed squares: sides versus areas
Join the midpoints of the sides of a square to get a new square tilted inside it. The area is cut in half each time, but what happens to the side length? Use the Pythagorean Theorem.
Look at the following fractal figure. We start with an equilateral triangle of area 64 sq in (Stage 1). In each subsequent step, in each of the triangles that remain…
Look at the following fractal figure. We start with an equilateral triangle of area 64 sq in (Stage 1). In each subsequent step, in each of the triangles that remain uncolored, we color the central triangle formed by the midpoints of its sides (inverted triangle), which has an area equal to $1/4$ of the triangle where it is located.
Considering the sequence formed by the total area of the new (orange) triangles colored at each stage (starting from Stage 2):
- a) Write the first three terms of the sequence.
- b) What is the common ratio of this progression?
- c) Calculate the sum of the areas of all the colored triangles if we repeated the process an infinite number of times.
We are designing an artistic logo based on a sequence of inscribed squares. We start with an outer square of side $L_1 = 20 \text{ cm}$. The second square is formed by…
We are designing an artistic logo based on a sequence of inscribed squares.
We start with an outer square of side $L_1 = 20 \text{ cm}$.
The second square is formed by joining the midpoints of the sides of the first one. The third square joins the midpoints of the second, and so on.
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Use the Pythagorean theorem to find the side of the second square ($L_2$) and the third ($L_3$).
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Determine the common ratio of the geometric progression formed by the areas of these squares.
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If we could draw infinite squares inwards following this pattern, what would be the sum of the areas of all the squares (the total colored area)?
A special rubber ball is dropped from a height of 2 feet. Due to energy loss, every time it bounces on the ground, it rises to only 80% of the height from which it…
A special rubber ball is dropped from a height of 2 feet. Due to energy loss, every time it bounces on the ground, it rises to only 80% of the height from which it previously fell.
This behavior generates a sequence of maximum heights reached after each bounce.
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Calculate the height the ball reaches after the fifth bounce.
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Imagine we let the ball bounce indefinitely until it stops. Calculate the total distance (downwards and upwards) that the ball will have travelled from when it was released until it stops completely. Be careful: the first drop is unique, but the bounces involve going up and down the same distance.
Fractals: the Sierpinski triangle
A fractal is built by repeating the same step again and again. The Sierpinski triangle starts with a solid triangle. At each step, every black triangle loses its central upside-down triangle, which has $\tfrac{1}{4}$ of its area. Each black triangle is replaced by 3 smaller ones.
Each step removes $\tfrac{1}{4}$ of the black area, so $\tfrac{3}{4}$ of it stays. The black area is multiplied by $r = \tfrac{3}{4}$ at every step. When a list starts at step 0, call the starting value $a_0$ and use $a_n = a_0 \cdot r^{\,n}$.
The Sierpinski Triangle is a famous fractal constructed from an equilateral triangle. The process is as follows: Step 0: We have a black equilateral triangle of area…
The Sierpinski Triangle is a famous fractal constructed from an equilateral triangle.
The process is as follows:
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Step 0: We have a black equilateral triangle of area $80 \text{ cm}^2$.
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Step 1: We cut out (remove) the inverted central triangle, whose area is $1/4$ of the previous triangle. We are left with 3 smaller black triangles.
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Step 2: From each of the 3 remaining black triangles, we again cut out the central triangle (which is $1/4$ of the area of its containing triangle).
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Write the sequence representing the total black area that remains in the figure at each step ($A_0, A_1, A_2, ...$).
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What is the common ratio $r$ of this decreasing geometric progression?
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Calculate what black area will remain after applying the process 5 times (Step 5).
Doubling, tripling and exponential growth
Geometric sequences with $r > 1$ model exponential growth: bacteria that double, followers that multiply, savings earning compound interest. It starts slowly, but then it outruns any "add the same amount" pattern.
Sometimes a problem gives a starting value at time 0. Then call it $a_0$ and use
$$a_n = a_0 \cdot r^{\,n}$$(Be careful: if the problem says $a_1$ is the first value, use $a_1 r^{n-1}$ instead. Read what $a_1$ means.)
We have a sheet of paper with an initial thickness of $0,1 \text{ mm}$. If we fold the sheet in half, the thickness doubles. If we fold it again, it doubles again.…
We have a sheet of paper with an initial thickness of $0,1 \text{ mm}$.
If we fold the sheet in half, the thickness doubles. If we fold it again, it doubles again.
Consider $a_1$ as the thickness after the first fold.
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Calculate the thickness of the sheet after 1, 2, 3 and 4 folds.
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What thickness will the stack of paper have after 10 folds? Express the result in millimeters.
A video by a famous streamer has gone viral. In the first hour it had 5 views. Due to the platform's algorithm, every hour that passes, the number of new views is…
A video by a famous streamer has gone viral. In the first hour it had 5 views. Due to the platform's algorithm, every hour that passes, the number of new views is multiplied by 3 compared to the previous hour.
- a) Write the first five terms of the sequence representing the new views per hour.
- b) What is the common ratio $r$ of this geometric progression?
- c) Write the general term $a_n$ that allows you to calculate the views in hour $n$.
- d) How many new views will the video have exactly in hour 8?
Percent growth and decay
When a quantity grows or shrinks by a percent each period, it is a geometric sequence. The ratio is the "growth factor":
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Increase by $p\%$: $r = 1 + \frac{p}{100}$. For example, +15% means $r = 1.15$.
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Decrease by $p\%$: $r = 1 - \frac{p}{100}$. For example, a ball that keeps 80% of its height has $r = 0.8$, and a price that drops 10% has $r = 0.9$.
A biologist is studying a colony of bacteria in a Petri dish. Initially, he counts 500 bacteria. He observes that the population increases by 20% every hour. a) Explain…
A biologist is studying a colony of bacteria in a Petri dish. Initially, he counts 500 bacteria. He observes that the population increases by 20% every hour.
- a) Explain why this growth corresponds to a geometric progression and determine the common ratio $r$.
- b) Write the general term $a_n$ that describes the number of bacteria after $n$ hours. (Consider $a_0 = 500$ as the start, or adjust the formula for $a_1$ as the amount after 1 hour).
- c) Approximately how many bacteria will there be after 12 hours?
A rubber ball is dropped from an initial height of 20 feet. On each bounce, the ball loses energy and only reaches 80% of the previous height ($r = 0,8$).
A rubber ball is dropped from an initial height of 20 feet. On each bounce, the ball loses energy and only reaches 80% of the previous height ($r = 0,8$).
A biologist is studying a very aggressive strain of bacteria in a Petri dish. Initially, he counts 500 bacteria. He observes that the population increases by 15% every…
A biologist is studying a very aggressive strain of bacteria in a Petri dish. Initially, he counts 500 bacteria. He observes that the population increases by 15% every hour.
For the experiment to be controllable, the population must not exceed 2,000,000 bacteria. Using a calculator and your knowledge of the general term of a geometric progression:
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Write the expression for the general term $a_n$ that describes the number of bacteria as a function of the hours elapsed $n$.
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Calculate exactly how many bacteria there will be after 24 hours.
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Determine the maximum number of full hours that can pass before the culture exceeds the safety limit of 2 million.
Arithmetic versus geometric
Compare $y = 2^x$ (geometric) with $y = 3x + 2$ (arithmetic). The linear one starts higher, but the exponential one eventually passes it and never looks back.
To compare totals instead of single values, remember that the sum of an arithmetic sequence is $S_n = \frac{n(a_1 + a_n)}{2}$, while the sum of a geometric one is $\frac{a_1(r^n - 1)}{r - 1}$.
A technology company offers two types of salary contract for a programmer position for 5 years: Contract A: Initial salary of $\$24,000…
A technology company offers two types of salary contract for a programmer position for 5 years:
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Contract A: Initial salary of $\$24,000$ per year, with a fixed increase of $\$2,000$ each year (Arithmetic Progression).
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Contract B: Initial salary of $\$22,000$ per year, with an 8% increase each year compared to the previous year (Geometric Progression).
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Write the general term $a_n$ for Contract A and $b_n$ for Contract B.
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Calculate the annual salary in the fifth year for both contracts. Which one pays more in that specific year?
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Calculate the total sum of money earned in the 5 years with each contract ($S_5$). Which contract is more beneficial in the long term (in the accumulated total)?
Back to the start: the viral video has $a_1 = 1$ and $r = 3$, so in minute 20 the number of new viewers is $a_{20} = 1 \cdot 3^{19} = 1{,}162{,}261{,}467$. That is more than a billion people from one share.