Further problems

F1

Consider the sequences below. Determine the 10th term, the sum of the $15$ first terms, and the infinite sum of the terms:

1. $4,4.2,4.41, ...$
2. $5,4.8, 4.6, ...$
3. $-5,4,-3.2, ...$
4. $-11,-6,-1, ...$
5. $-\frac{2}{3}, \frac{1}{3} , -\frac{1}{6}, ...$

F2

Determine the sums

1. $0.25+0.75+2.25+...+44286.75$
2. $2.1+3.2+4.3+...+18.6$

F3

A geometric object grows as follows (see sketch below). In generation $1$ it is a square of length $2$. In each subsequent generation, more squares are added, with the side length shrinking by $60\%$ each time.

1. In which generation does the attached square have an area smaller than $0.001$ for the first time?
2. Determine the total area of the object in the $10$th generation.
3. Determine the total area of the object after infinitely many iterations.

F4

The object below consists of infinitely many equilateral triangles, the first triangle having side length $1$. The side length is halved after every iteration.

1. Determine the length of the fat spiral $a_1, a_2, ...$
2. Determine the total area $s_\infty$ of all triangles.
3. After how many triangles is the area $s_n$ $99.9\%$ of the total area $s_\infty$?

F5

Snowflake or Koch-curve. An equilateral triangle has side length 1 (generation 1). Further equilateral triangles are attached as shown below.

1. Determine the perimeter of the object after an infinite number of iterations.

2. Determine the area of the object after an infinite number of iterations.

Solution