Geometric sequences and series

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 $a_6$.

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 $r = 2$.

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.

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 $a_8 = 128$.

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 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 the end of year $n$. How much money will be in the account after $8$ years if no money is withdrawn?

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.

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 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.

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.