Exercises: Geometric progressions

Write 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 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 $a_1$.

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.

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

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 after $24$ hours? Express the result using powers.

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.

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.

$\$5000$ 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 How much money will be in the account after $8$ years if nothing is withdrawn?

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.

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.

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 $n \to \infty$.

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.

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.

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.

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 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.

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.

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 $b_n$.