Drop a basketball from 10 feet. Each time it hits the floor, it bounces back up to 60% of its previous height. In theory it bounces infinitely many times. Does it travel an infinite distance? By the end of this page you'll know. First we need the tools. Some problems call these sequences geometric progressions. It's the same thing.
The explicit formula and the common ratio
Why the exponent is $n-1$: to get from $a_1$ to $a_n$ you multiply by $r$ exactly $n-1$ times. To get from $a_k$ to $a_n$ you multiply $n-k$ times.
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Given the sequence defined by the general term $a_n = 5 \cdot (-2)^{n-1}$, calculate the value of the terms $a_3$ and $a_6$.
Given the sequence defined by the general term $a_n = 5 \cdot (-2)^{n-1}$, calculate the value of the terms $a_3$ and $a_6$.
Determine whether the following sequence is a geometric progression. If so, find its common ratio and its general term:…
Determine whether the following sequence is a geometric progression. If so, find its common ratio and its general term: $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, \dots$
In a geometric progression of positive terms, it is known that $a_2 = 6$ and $a_4 = 54$. Calculate the value of the common ratio $r$ and the first term $a_1$.
In a geometric progression of positive terms, it is known that $a_2 = 6$ and $a_4 = 54$. Calculate the value of the common ratio $r$ and the first term $a_1$.
In a geometric progression it is known that $a_2 = 12$ and $a_5 = 324$. Calculate the common ratio of the progression and the sum of the first six terms.
In a geometric progression it is known that $a_2 = 12$ and $a_5 = 324$. Calculate the common ratio of the progression and the sum of the first six terms.
According to the legend of the inventor of chess, the king offered him one grain of wheat on the first square, two on the second, four on the third, and so on, doubling…
According to the legend of the inventor of chess, the king offered him one grain of wheat on the first square, two on the second, four on the third, and so on, doubling the amount on each square until reaching number $64$. Express the total amount of wheat grains on the board using the formula for the sum of a geometric progression. If one gram of wheat contains about $20$ grains and a ton is $10^6$ grams, estimate the order of magnitude of the total tons of wheat.
Sometimes you need to multiply the terms instead of adding them. Look at $1, 3, 9, 27, 81$. Terms the same distance from the two ends have the same product: $1\cdot 81 = 3\cdot 27 = 81$. The middle term is $9=\sqrt{81}$.
A useful consequence: if $i+j=k+l$, then $a_i\cdot a_j = a_k\cdot a_l$.
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Calculate the product of the first six terms of a geometric progression that has $a_1 = 2$ as its first term and $r = 2$ as its common ratio.
Calculate the product of the first six terms of a geometric progression that has $a_1 = 2$ as its first term and $r = 2$ as its common ratio.
Prove that the product of the first $n$ terms of a geometric progression with positive terms is given by the formula $P_n = \sqrt{(a_1 \cdot a_n)^n}$. Use this result to…
Prove that the product of the first $n$ terms of a geometric progression with positive terms is given by the formula $P_n = \sqrt{(a_1 \cdot a_n)^n}$. Use this result to calculate the product of the first $10$ terms of a geometric progression in which $a_3 = 4$ and $a_8 = 128$.
The sum to infinity of a geometric progression is $18$ and its first term is $6$. Determine the common ratio of the progression and the value of the fourth term.
The sum to infinity of a geometric progression is $18$ and its first term is $6$. Determine the common ratio of the progression and the value of the fourth term.
A percent change is a multiplication, so repeated percent changes make a geometric sequence. Convert the percent into a ratio:
Situation
Ratio $r$
Increase of $p\%$ per period
$r = 1 + p$ (7% growth: $r=1.07$)
Decrease of $p\%$ per period
$r = 1 - p$ (12% decay: $r=0.88$)
Doubles / triples per period
$r=2$ / $r=3$
If the starting amount is $a_0$ (time $0$), the amount after $n$ periods is $a_n=a_0 r^n$. If you call the starting amount $a_1$, use $a_n = a_1r^{n-1}$ instead. Both are fine. Just be consistent.
For savings with compound interest, the amount after $t$ years is $A=P(1+r)^t$, where $P$ is the principal and $r$ the annual rate.
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A piece of industrial equipment costs $\$24\,000$ and depreciates by $12\%…
A piece of industrial equipment costs $\$24\,000$ and depreciates by $12\%$ each year compared to the value of the previous year. Write the general term of the sequence that models its value and calculate how much it will be worth after $6$ years.
In a laboratory culture, a population of bacteria triples every $4$ hours. If $100$ bacteria are initially introduced into the dish, how many bacteria will there be…
In a laboratory culture, a population of bacteria triples every $4$ hours. If $100$ bacteria are initially introduced into the dish, how many bacteria will there be after $24$ hours? Express the result using powers.
$\$5000$ are deposited in a financial institution at a compound annual interest of $3,5\%…
$\$5000$ are deposited in a financial institution at a compound annual interest of $3,5\%$. Write the general expression for the accumulated capital at the end of year $n$. How much money will be in the account after $8$ years if nothing is withdrawn?
Infinite processes: bounces, squares, and paradoxes
Many "forever" situations have a finite answer. Find the first term, find $r$, check $|r|<1$, then use $S_\infty$.
Back to the opening: the ball bounces infinitely often, yet it travels only 40 ft in total, because the bounce heights form a geometric series with $|r|<1$.
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An elastic ball is dropped from an initial height of $5$ feet. On each bounce, the ball rebounds reaching $80\%$ of the height of the previous bounce. Set up the…
An elastic ball is dropped from an initial height of $5$ feet. On each bounce, the ball rebounds reaching $80\%$ of the height of the previous bounce. Set up the sequence of the heights reached and calculate the total distance travelled by the ball (adding the descents and ascents) until it theoretically stops.
There is a square of side $10$ inches. By joining the midpoints of its sides, a second square is formed. By joining the midpoints of the sides of this second square, a…
There is a square of side $10$ inches. By joining the midpoints of its sides, a second square is formed. By joining the midpoints of the sides of this second square, a third square is formed, and so on. Calculate the sum of the areas of the infinite squares formed by this process.
Achilles and the tortoise are having a race. Achilles runs at $10$ ft/s and the tortoise at $1$ ft/s. The tortoise starts with a head start of $100$ feet. When Achilles…
Achilles and the tortoise are having a race. Achilles runs at $10$ ft/s and the tortoise at $1$ ft/s. The tortoise starts with a head start of $100$ feet. When Achilles covers those $100$ feet, the tortoise has advanced $10$ feet. When Achilles covers those $10$ feet, the tortoise advances $1$ foot, and so on. Set up the total distance Achilles runs as the sum to infinity of a geometric progression and prove that the result is finite.
To prove that a sequence is geometric, show that $\dfrac{a_{n+1}}{a_n}$ is the same constant for every $n$. In figure problems, describe what happens in one step: how many pieces, and how big each one is.
Two facts help a lot.
If every length is multiplied by $k$, every area is multiplied by $k^2$. A shape scaled by $\tfrac13$ has $\tfrac19$ of the area.
The limit of $a_n=a_1r^{n-1}$ as $n\to\infty$: if $|r|<1$ it is $0$. If $r>1$ and $a1>0$, the terms grow without bound (the limit is $\infty$).
Now a shape where the length does the opposite. In the Koch snowflake, you don't remove the middle third of each segment. You replace it with a point.
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The curve known as the "Koch snowflake" is constructed from an equilateral triangle of side $L$. In each step, the middle third of each segment is replaced by two…
The curve known as the "Koch snowflake" is constructed from an equilateral triangle of side $L$. In each step, the middle third of each segment is replaced by two segments of equal length that form an outward point. Let $P_n$ be the perimeter of the figure in step $n$ (where $P_0 = 3L$). Prove that $P_n$ is a geometric progression, find its general term and calculate the limit of the perimeter as $n \to \infty$.
A circle is inscribed in a square of side $L = 20$ inches. Inside this circle a new square is inscribed, in which another circle is inscribed, and the process is…
A circle is inscribed in a square of side $L = 20$ inches. Inside this circle a new square is inscribed, in which another circle is inscribed, and the process is repeated indefinitely. Find the expression for the area of the nth circle and calculate the sum of the areas of the infinite circles formed.
Three numbers $a, b, c$ are consecutive terms of a geometric sequence when the ratios match: $\dfrac{b}{a}=\dfrac{c}{b}$. Cross-multiplying gives a rule that turns "is geometric" into an equation.
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Find the value of the real number $x$ so that the terms $x-1$, $2x$, and $5x+4$ form, in that order, a geometric progression.
Find the value of the real number $x$ so that the terms $x-1$, $2x$, and $5x+4$ form, in that order, a geometric progression.
In a geometric progression of positive terms, the sum of the first two terms is $15$ and the sum of the third and fourth terms is $60$. Set up the corresponding system…
In a geometric progression of positive terms, the sum of the first two terms is $15$ and the sum of the third and fourth terms is $60$. Set up the corresponding system of equations to determine the first term and the common ratio of the progression.
The interior angles of a convex quadrilateral are in a geometric progression with common ratio $2$. Calculate the measure of each of the four angles. Would it be…
The interior angles of a convex quadrilateral are in a geometric progression with common ratio $2$. Calculate the measure of each of the four angles. Would it be possible to construct a convex pentagon whose interior angles form a geometric progression with common ratio $2$? Justify your answer.
Taking logarithms turns multiplication into addition. So taking logs of a geometric sequence should turn it into an arithmetic one.
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Let $a_n$ be a geometric progression of strictly positive terms with first term $a_1$ and common ratio $r$. Prove that the sequence defined by $b_n = \log(a_n)$ is an…
Let $a_n$ be a geometric progression of strictly positive terms with first term $a_1$ and common ratio $r$. Prove that the sequence defined by $b_n = \log(a_n)$ is an arithmetic progression. If $a_1 = 10$ and $r = 100$, calculate the sum of the first $20$ terms of the progression $b_n$.