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Solved problems for the chapter:

Infinite geometric series

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Infinite sequence

1. Near point

2. Infinite sequence limit

2.1. Convergence sequence

2.2. Divergent sequence

2.3. Limit of a sequence

3. Evaluation of limits

3.1. Euler's number

4. Forum

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Geometric sequence

1. General term of a geometric sequence

1.1. Common ratio of a geometric sequence

2. Geometric mean

2.1. Relationship between geometric and arithmetic mean

2.1.1. Numerical Proof

2.1.2. Graphical proof

3. Forum

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Infinite geometric series

1. Infinite sequence

1.1. Sum of an infinite sequence

1.2. Convergence of an infinite geometric sequence

2. Forum

↓ Show theory [3] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:infinite_geometric_series/5417/

Example 1.

The square with side is divided into four equal squares with side . Thus we continue the process of divisions to infinity. Calculate the sum of the areas by adding one of the squares of each division to the square with side .

Example 2.

The first three terms of the geometric sequence are . Calculate the common ratio and the fourth term of this sequence. Also calculate the sum of the corresponding geometric series . In the fraction, eliminate the root in the denominator.

Evaluate:

Obtain the first three terms of an infinite geometric sequence with the following data:

10.

Obtain the first three terms of an infinite geometric series with the following data:

11.

The sum of the first two terms of a geometric sequence is 35. If the sum to infinity of the sequence is 80, obtain the sequence.

12.

Calculate the total length of the lines in the figure below.

13.

The sequence of areas and the sequence of perimeters of the equilateral triangles in the figure below form an infinite geometric sequence each. Find the sum of the perimeters and the sum of the areas of all the equilateral triangles.