Geometric Progressions
Yaritza puts $\text{RD}\$5{,}000$ into a credit union that pays compound interest every year. A bacterium triples, a ball bounces lower and lower each time, a piece of equipment loses value year after year. In all of these cases, each term comes from multiplying the previous one by a fixed number. On this page you will learn to describe them, to add them up (even with infinitely many terms), and to solve problems. At the end you will see why Achilles does catch the tortoise.
General Term and Common Ratio
Why? To get from $a_1$ to $a_n$ you multiply by $r$ exactly $n-1$ times: $a_2=a_1r$, $a_3=a_1r^2$, $a_4=a_1r^3$… From $a_k$ to $a_n$ there are $n-k$ multiplications.
To find out whether a sequence is geometric, divide each term by the previous one: if the quotient $\dfrac{a_{n+1}}{a_n}$ is always the same, that quotient is the common ratio.
Write down the first five terms and find the expression for the general term of a geometric progression, knowing that its first term is $a_1 = 3$ and its common ratio is…
Write down the first five terms and find the expression for the general term of a geometric progression, knowing that its first term is $a_1 = 3$ and its common ratio is $r = 2$.
Given the sequence defined by the general term $a_n = 5 \times (-2)^{n-1}$, calculate the value of the terms $a_3$ and $a_6$.
Given the sequence defined by the general term $a_n = 5 \times (-2)^{n-1}$, 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 with 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 with 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 $a_1$.
Product of the First Terms
Sometimes you are asked to multiply the first $n$ terms instead of adding them. There is a trick: in a geometric progression, terms that are equally far from the two ends have the same product.
Calculate the product of the first six terms of a geometric progression that has first term $a_1 = 2$ and common ratio $r = 2$.
Calculate the product of the first six terms of a geometric progression that has first term $a_1 = 2$ and common ratio $r = 2$.
Prove that the product of the first $n$ terms of a geometric progression with positive terms is given by the formula $P_n = \sqrt{(a_1 \times a_n)^n}$. Use this result…
Prove that the product of the first $n$ terms of a geometric progression with positive terms is given by the formula $P_n = \sqrt{(a_1 \times a_n)^n}$. 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 of the First $n$ Terms
Proof. We write $S_n = a_1 + a_1r + a_1r^2+\dots+a_1r^{n-1}$. Multiplying by $r$, each term shifts to the next one: $rS_n = a_1r + a_1r^2+\dots+a_1r^{n}$. Subtracting, all the middle terms cancel: $rS_n - S_n = a_1r^n - a_1$. Factoring, $S_n(r-1)=a_1(r^n-1)$, and the formula follows.
Calculate the sum of the first $8$ terms of a geometric progression in which the first term is $a_1 = 4$ and the common ratio is $r = 3$.
Calculate the sum of the first $8$ terms of a geometric progression in which the first term is $a_1 = 4$ and the common ratio is $r = 3$.
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 up to square number $64$. Express the total number of grains of wheat on the board using the formula for the sum of a geometric progression. If one gramme of wheat contains about $20$ grains and one metric ton is $10^6$ grammes, estimate the order of magnitude of the total metric tons of wheat.
Infinite Sum When $|r| < 1$
If $-1 Find the sum to infinity of a geometric progression whose first term is $a_1 = 10$ and whose common ratio is $r = \frac{1}{5}$. Express the recurring decimal $0.4444\dots$ as the sum of the infinite terms of a geometric progression and obtain its generating fraction. 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.Find the sum to infinity of a geometric progression whose first term is $a_1 = 10$ and whose common ratio is $r = \frac{1}{5}$.
Express the recurring decimal $0.4444\dots$ as the sum of the infinite terms of a geometric progression and obtain its generating fraction.
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.
Finance: Compound Interest, Present Value, and Depreciation
If a principal $C$ grows by a percentage $i$ (in decimal form) every period, the principal is multiplied by $(1+i)$ each time. It is a geometric progression with ratio $1+i$. If instead it loses a percentage $d$, the ratio is $1-d$.
An item of industrial equipment costs $1\,440\,000$ pesos and depreciates by $12\%$ each year relative to the previous year's value. Write down the general term of the…
An item of industrial equipment costs $1\,440\,000$ pesos and depreciates by $12\%$ each year relative to the previous year's value. Write down the general term of the sequence that models its value and calculate how much it will be worth after $6$ years.
$300\,000$ pesos are deposited in a financial institution at an annual compound interest rate of $3.5\%$. Write down the general expression for the capital accumulated…
$300\,000$ pesos are deposited in a financial institution at an annual compound interest rate of $3.5\%$. Write down the general expression for the capital accumulated at the end of year $n$. How much money will be in the account after $8$ years if no money is withdrawn?
Loans with Fixed Payments
When you take out a loan at a commercial bank or a credit union, you make equal payments every month. How is the payment calculated? The key idea: the loan received today equals the sum of the present values of all the payments.
If the rate per period is $i$ and $n$ payments of amount $c$ are made at the end of each period, the present value of payment $k$ is $\dfrac{c}{(1+i)^k}$. Those values form a geometric progression with $a_1=\dfrac{c}{1+i}$ and ratio $r=\dfrac{1}{1+i}$.
Problems with Geometric Progressions
Bacteria populations, bounce heights, repeating figures, equations to find $a_1$ and $r$: identify $a_1$ and $r$, and decide whether the question is about a term, a finite sum, or an infinite sum.
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.
Find the value of the real number $x$ such that the terms $x-1$, $2x$, and $5x+4$ form, in that order, a geometric progression.
Find the value of the real number $x$ such that the terms $x-1$, $2x$, and $5x+4$ form, in that order, a geometric progression.
An elastic ball is dropped from an initial height of $5$ meters. On each bounce, the ball reaches $80\%$ of the height of the previous bounce. Set up the sequence of…
An elastic ball is dropped from an initial height of $5$ meters. On each bounce, the ball reaches $80\%$ of the height of the previous bounce. Set up the sequence of heights reached and calculate the total distance traveled by the ball (adding the descents and ascents) until it theoretically comes to a stop.
A square has side length $10$ cm. Joining the midpoints of its sides forms a second square. Joining the midpoints of the sides of this second square forms a third…
A square has side length $10$ cm. Joining the midpoints of its sides forms a second square. Joining the midpoints of the sides of this second square forms a third square, and so on. Calculate the sum of the areas of the infinite squares formed by this process.
The curve known as the "Koch snowflake" is constructed starting from an equilateral triangle with side length $L$. At each step, the middle third of each segment is…
The curve known as the "Koch snowflake" is constructed starting from an equilateral triangle with side length $L$. At 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 perimetre of the figure at step $n$ (where $P_0 = 3L$). Prove that $P_n$ is a geometric progression, find its general term, and calculate the limit of the perimetre as $n \to \infty$.
In a geometric progression with 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 with 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.
Achilles and the tortoise run a race. Achilles runs at $10$ m/s and the tortoise at $1$ m/s. The tortoise starts with a head start of $100$ meters. When Achilles travels…
Achilles and the tortoise run a race. Achilles runs at $10$ m/s and the tortoise at $1$ m/s. The tortoise starts with a head start of $100$ meters. When Achilles travels those $100$ meters, the tortoise has advanced $10$ meters. When Achilles travels those $10$ meters, the tortoise advances $1$ meter, and so on. Set up the total distance traveled by Achilles as the sum of the infinite terms of a geometric progression and prove that the result is finite.
The interior angles of a quadrilateral are in a geometric progression with common ratio $2$. Calculate the measure of each of the four angles. Would it be possible to…
The interior angles of a 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.
A circle is inscribed in a square with side length $L = 20$ cm. Inside this circle, a new square is inscribed, in which another circle is inscribed, and the process…
A circle is inscribed in a square with side length $L = 20$ cm. Inside this circle, a new square is inscribed, in which another circle is inscribed, and the process repeats indefinitely. Find the expression for the area of the nth circle and calculate the sum of the areas of the infinite circles formed.
Let $a_n$ be a geometric progression with strictly positive terms, 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 with strictly positive terms, 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 $b_n$.
Back to the beginning: Yaritza's principal grows like a geometric progression with ratio $1+i$, and in Achilles's race the infinitely many distances form a progression with ratio $\frac{1}{10}<1$: their sum $S_\infty=\frac{a_1}{1-r}$ is finite, so Achilles catches the tortoise.