1Question 1
Determine whether the infinite geometric series $2 + 1 + 0.5 + 0.25 + \dots$ converges. If it does, find the sum.
2Question 2
Express $0.\overline{7}$ (0.777...) as a fraction in simplest form.
3Question 3
Find the sum of the infinite series $\sum_{n=1}^{\infty} 5(0.4)^{n-1}$.
4Question 4
A geometric series has first term 12 and infinite sum 16. Find the common ratio $r$.
5Question 5
Express $0.\overline{12}$ (0.121212...) as a fraction in simplest form.
6Question 6
Does the series $3 - 1.5 + 0.75 - 0.375 + \dots$ converge? If so, find its sum.
7Question 7
A ball is dropped from a height of 80 cm. Each bounce reaches 70% of the previous height. Find the total vertical distance traveled if it bounces infinitely many times.
8Question 8
Find the value of $x$ if $\sum_{k=0}^{\infty} 2^{-k} = x$.
9Question 9
A shrub is 110 cm tall. It grows 10 cm in the first year, then each year it grows half of what it grew the previous year. What is the maximum height the shrub will reach?
10Question 10
Determine whether each of the following series converges or diverges: (a) $\sum_{n=1}^{\infty} 3(1.2)^{n-1}$, (b) $\sum_{n=1}^{\infty} 4(0.8)^{n-1}$, (c) $\sum_{n=1}^{\infty} 5(-0.9)^{n-1}$.