Geometric progressions
A bacterium triples, a ball bounces to a lower height each time, an investment earns compound interest. In every case, each term comes from multiplying the previous one by a fixed number. You already saw this in 3rd ESO: now you are going to combine it with logarithms, equations and the paradox of Achilles and the tortoise.
Revision: general term and 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 $a_n = 5 \cdot (-2)^{n-1}$, 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 $a_6$.
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 $a_1$.
The sum of the first $n$ terms
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 $r = 3$.
An item of industrial equipment costs $24\,000$ euros 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 $24\,000$ euros 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.
$5000$ euros 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…
$5000$ euros 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 How much money will be in the account after $8$ years if no money is withdrawn?
The 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. 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.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.
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…
Problems with geometric progressions
Depreciation, bacteria populations, bounce heights, fractals: identify $a_1$ and $r$, and decide whether the question is about a specific 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.
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 $100$ metres. 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 $100$ metres. When Achilles travels those $100$ metres, the tortoise has advanced $10$ metres. When Achilles travels those $10$ metres, the tortoise advances $1$ metre, and so on. Set up the total distance travelled by Achilles as the sum of the infinite terms of a geometric progression and prove that the result is finite.
Back to the beginning: In the paradox of Achilles and the tortoise, the infinitely many distances covered form a geometric progression with common ratio $r < 1$: their total sum $S_\infty = \frac{a_1}{1-r}$ is finite, so Achilles does catch up with the tortoise in a perfectly finite distance and time.