Exercises: Geometric progressions

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 $a_6$.

Determine whether the following sequence is a geometric progression. If so, find its common ratio and its general term: $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, \dots$

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 $r = 3$.

Find the sum to infinity of a geometric progression whose first term is $a_1 = 10$ and whose common ratio is $r = \frac{1}{5}$.

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 $a_1$.

Calculate the product of the first six terms of a geometric progression that has first term $a_1 = 2$ and common ratio $r = 2$.

Express the recurring decimal $0,4444\dots$ as the sum of the infinite terms of a geometric progression and obtain its generating fraction.

Lucía is trying to calculate the fifth term ($a_5$) of a geometric progression whose first term is $a_1 = 8$ and whose common ratio is $r = 0,5$. In her notebook, she writes the following working:

1. a) Without performing analytical calculations, explain a conceptual reason why the result $256$ is inconsistent with the initial data ($a_1 = 8$ and $r = 0,5$).
2. b) Identify the mistake made by Lucía in the application of the order of operations or the general formula $a_n = a_1 \cdot r^{n-1}$.
3. c) Correct the procedure and determine the exact value of $a_5$ in both fractional and decimal form.

Consider the geometric progression with first term $a_1 = 18$ and common ratio $r = \frac{1}{3}$.

1. a) Write its first four terms. Based on them, make a reasoned estimate: between which two consecutive integers do you think the sum of its infinite terms ($S_\infty$) will lie? Explain your intuition before using any formula.
2. b) Apply the theoretical formula for the infinite sum $S_\infty = \frac{a_1}{1-r}$ to calculate its exact value and check whether your initial estimate was correct.
3. c) Calculate the finite sum of the first four terms ($S_4$) and determine what percentage of the total infinite sum ($S_\infty$) has already been reached by summing only those first four terms.

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.

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.

An elastic ball is dropped from an initial height of $5$ metres. 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 travelled by the ball (adding the descents and ascents) until it theoretically comes to a stop.

$5000$ 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?

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.

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 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.

According to INE reports on the demographic challenge in inland Spain, a municipality in the province of Teruel had a registered census of $3\,125$ inhabitants on January 1, 2020. Due to population aging and the lack of generational replacement, the census constantly decreases annually by $4\%$ relative to the immediately preceding year.

1. a) Model the municipal population as of January 1 of each successive year using a geometric progression. Justify the value of the first term $a_1$ (year 2020) and deduce the common ratio $r$.
2. b) Write the general term $a_n$ of the sequence and calculate the projected population for January 1, 2026.
3. c) Legislation considers a town to be at "extreme risk of depopulation" if its census drops below $2\,000$ inhabitants. Set up an inequality and solve it using logarithms to determine starting from which year this municipality will enter that state if the trend continues.

To calibrate a physical sensor, it is necessary to reconstruct a geometric progression with strictly positive real terms. Only two overall records are known about it: the sum of its first two terms is $a_1 + a_2 = 24$, and the sum of its infinite terms converges to $S_\infty = 32$.

1. a) Set up a system of equations in terms of the first term $a_1$ and the common ratio $r$ using the formal expressions for $a_2$ and $S_\infty$.
2. b) Solve the system and show that two possible algebraic values arise for the common ratio $r$. Justify in detail why one of them must be discarded based on the conditions of the statement.
3. c) From the valid solution, formulate the general term $a_n$ and calculate the exact value of the sum of the first six terms ($S_6$).

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 of equations to determine the first term and the common ratio of the progression.

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 tonne is $10^6$ grammes, estimate the order of magnitude of the total tonnes of wheat.

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.

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 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 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 arithmetic progression. If $a_1 = 10$ and $r = 100$, calculate the sum of the first $20$ terms of the progression $b_n$.

To find the irreducible generating fraction of the mixed recurring decimal $1{,}2\widehat{45} = 1{,}2454545\dots$, two 1st-year Bachillerato students advocate different methods:

• Method 1 (Cancellation by powers of 10): Multiply the number $x = 1{,}2\widehat{45}$ by $1000$ and by $10$, subtract both equations term by term ($1000x - 10x$) to eliminate the infinite recurring tail, and solve for $x$.

• Method 2 (Sum of an infinite geometric series): Decompose the number into its non-recurring part plus an infinite sum:

Identify the geometric progression starting at the term $\frac{45}{10^3}$, find its sum $S_\infty$ using its formula, and add it to the initial decimal fraction.

1. a) Fully work out both methods until reaching the same irreducible fraction in both cases.
2. b) Critically compare the two procedures: which one is more agile for routine calculation, and why does Method 2 provide the true rigorous mathematical foundation (theory of limits and convergent series) that validates the subtraction of infinite decimal places that Method 1 applies without justification?

In a hospital treatment, a patient is administered a fixed dose of $120\text{ mg}$ of a drug every $12$ hours. The body metabolizes the substance in such a way that, when $12$ hours have elapsed after a dose (just before the next one), only $25\%$ of the amount of drug that was present immediately after that dose remains in the bloodstream.

1. a) Model the amount of antibiotic present in the blood immediately after the 1st, 2nd, and 3rd dose. Deduce with reasoning the expression for the accumulated amount $C_n$ just after the $n$-th dose as the sum of the terms of a geometric progression.
2. b) The toxicity threshold of the medication is set at $170\text{ mg}$. In long-term treatments (more than three months), healthcare personnel fear that the drug may accumulate uncontrollably and exceed that threshold. Prove mathematically, using the limit of the accumulated sum as $n \to \infty$, why that risk does not exist, and calculate the theoretical maximum concentration that the patient will never exceed.
3. c) If, after virtually reaching steady state, the patient accidentally forgets a dose and receives the next one $24$ hours after the previous one, calculate the amount of drug that will remain in their blood just before taking the medication again and after receiving the new dose.