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mathematics UK Year 11 Mathematics (KS4)
 
Solved problems for the chapter:
 

Geometric Sequences

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Geometric sequence
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Detailed solution steps are available at:
 
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Example 1.

The fifth term of the geometric sequence is eight times the second term, and the product of the second and fourth terms is . Calculate the first term and the common ratio .

    
Example 2.

Determine whether the values of the following expressions are consecutive terms of a geometric sequence. Write down the common ratio of the sequence:


    
3.

Determine whether the values of the following number expressions are consecutive terms of a geometric sequence. Write down the common ratio of the sequence.


    
4.

The number is written as the sum of the first three terms of the arithmetic sequence. If we increase the third term of this sequence by , we get three consecutive terms of a geometric sequence. Write down the arithmetic sequence members and the geometric sequence members.

    
5.

Three numbers whose sum is form a geometric sequence. If we increase the second term of the sequence by and decrease the third by , we obtain the consecutive terms of the arithmetic sequence. Write down these terms of the geometric sequence.

    
6.

In a square with side a unit we draw a square with vertices in the midpoints of the sides of the previous square, in this square we draw a square with vertices in the midpoints of the sides of the previous square, etc. Calculate the lengths of the sides of the first three squares and describe the lengths of the side of the squares with the sequence.

    
7.

Draw a circle of radius , inscribe in it an equilateral triangle then inscribe a circle in the triangle. Repeat the processes for a number of times. Describe the size of the areas of the obtained triangles with the sequence.

    
8.

A ball bounces off the horizontal floor after the first bounce , after the second … The heights to which the ball bounces form a geometric sequence. How much does the ball bounces after the third rebound and how much after the fifth? Describe the height of the ball's bounce with the general term of the sequence. After which rebound does the ball bounce less than ?

    

9.

Write down the common ratio and the general term of the geometric sequence and write down its properties:


    
10.

Write down the common ratio and the general term of the geometric sequence and write down its properties:


    
11.

Write down the common ratio and the general term of the geometric sequence and write down its properties:


    
12.

Write down the common ratio and the general term of the geometric sequence and write down its properties:


    
13.

Write down the common ratio and the general term of the geometric sequence and write down its properties:


    
14.

We write down four natural numbers for which the first three form an arithmetic sequence and the first, second, and fourth numbers form a geometric sequence. The sum of the first and third numbers is , and the sum of the second and fourth is . Find the natural numbers.