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mathematics UK Year 10 Mathematics (KS4)
 
Solved problems for the chapter:
 

Geometric sequence

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:geometric_sequence/5346/


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.

Write down the properties of the sequence given by the general term .

    
3.

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


    
4.

The figure shows a graph of the geometric sequence. Write down the general term of the sequence. How many terms of the sequence are greater than ?


    
5.

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


    
6.

Write down the properties of the sequences given by the general term .

    
7.

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.

    
8.

Write the first five terms of the geometric sequence if .

    

9.

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.

    
10.

Write down the properties of the sequences given by the general term .

    
11.

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.

    
12.

Write the first six terms of the sequence with the general term and draw its graph.

    
13.

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.

    
14.

Write down the properties of the sequences given by the general term .

    
15.

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 ?

    
16.

Write the first five terms of the geometric sequence if .

    
17.

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


    
18.

Write the first six terms of the sequence with the general term and draw its graph.

    

19.

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


    
20.

Write the first five terms of the geometric sequence if .

    
21.

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


    
22.

Write down the first six terms of the sequence given by the general term .

    
23.

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


    
24.

Mike drew a sequence of triangles. At each step, he counted how many of the smallest triangles in the picture.




There are 1 in the first picture, 4 in the second, and 16 of the smallest triangles in the third. Calculate how many smallest triangles there would be in the fifth figure.

    
25.

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


    
26.

Write the first five terms of the geometric sequence if .

    
27.

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.

    
28.

Write the first six terms of the sequences given by the general term .

    

29.

For the given data for geometric sequence, find the unknown quantities:


    
30.

Write the first six terms of the sequences given by the general terms .

    
31.

For the given data for geometric sequences, find the unknown quantities:


    
32.

Write the first six terms of the sequence given by the general term .

    
33.

For the given data for geometric sequences, find the unknown quantities:


    
34.

Write the first six terms of the sequences given by the general term .

    
35.

For the given data for geometric sequences, find the unknown quantities:


    
36.

Write the first six members of the sequences given by the general article .

    
37.

Write the first six terms of the sequences given by the general terms .

    
38.

Write the first nine terms of the sequence given by the general term and draw a graph of the sequence.

    

39.

Show that the numbers form a geometric sequence.

    
40.

For the given sequence find the general term.

    
41.

For the given sequence find the general term.

    
42.

For the given sequence find the general term.

    
43.

For the given sequence find the general term.

    
44.

For the given sequence find the general term.

    
45.

For the given sequence find the general term.

    
46.

For the given sequence find the general term.

    
47.

The figure shows the graph of a sequence. Write down the expression for the general term.


    
48.

Write down the properties of the sequences given by the general term .

    

49.

Show that are the first three terms of the descending arithmetic sequence. Write down the difference of the sequence and calculate the fifth term.

    
50.

Write down the general term of the sequence . Which term of the sequence is equal to ?

    
51.

Calculate the first term and the common ratio of the ascending geometric sequence in which:






Write the first four terms of this geometric sequence.

    
52.

Write down the first terms of the sequence with general term . How many terms lie in and how many in ?

    
53.

Write the first six term of the sequences given by the general terms .

    
54.

Write the first term of the sequence and draw the points they represent on the number line. How many terms are smaller than ?

    
55.

Spheres are stacked as shown in the figure below. The numbers of spheres on each level that form a square make up a sequence. The sequence is described so that for , (there is one sphere at the top), for , the number of spheres below the upper sphere in the second row (there are four spheres there).


Write down the formula used to calculate the number of spheres in the nth row (looking from top to bottom). How many of all the balls are in the pile in the figure?


    
56.

Calculate for which real numbers x the values of the given expressions are consecutive terms of a geometric sequence.


    
57.

Calculate for which real numbers x the values of the given expressions are consecutive terms of a geometric sequence.


    
58.

Calculate for which real numbers x the values of the given expressions are consecutive terms of a geometric sequence.


    

59.

Calculate for which real numbers x the values of the given expressions are consecutive terms of a geometric sequence.


    
60.

The sequence is given:




  • Write down the first four terms of the sequence and calculate their sum.


  • Which term of the sequence is .


  • What are the upper bound and the lower bound of the given sequence? Justify your answers.

    
61.

Calculate for which real numbers x the values of the given expressions are consecutive terms of a geometric sequence.


    
62.

Write the general term of the geometric sequence given by:


    
63.

Write the general term of the geometric sequence given by:


    
64.

Write the general term of the geometric sequence given by:


    
65.

In the geometric sequence, the sixth term is and the common ratio is . What is the ninth term of the sequence?

    
66.

Find the fifth, tenth and general terms of the geometric sequence

   
67.

Find the fourth, eighth, and general terms of the geometric sequence . Are the numbers terms of the given sequence?

    
68.

Find the fourth, eighth, and general terms of the geometric sequence

   

69.

Let , , , ... be the positive terms of a geometric sequence.


  • Calculate the terms and and write down the general term of this sequence.


  • Calculate between which consecutive terms of the sequence is the number . Write the answer.


  • Prove that the sum of is equal to .


  • Calculate the sum of type .

    
70.

What is all three-digit natural numbers that give the remainder 2 when divided by 25?

    
71.

Calculate the first three terms of the sequence with the general term . Which term in this sequence is equal to ? Prove that the sequence is descending .

    
72.

Sequence with general term is given. Determine the properties of the sequence and also prove them.

    
73.

Show that the given sequence is bounded above and below.

    
74.

The numbers are the first three terms of the geometric sequence. Calculate x and write the fifth term of this sequence.

    
75.

In a family, the ages of the son, daughter, and father are successive terms of a geometric sequence with a common ratio of . Their total age is . Calculate the age of the father.

    
76.

Insert five numbers between the numbers 7 and 448 so that we get:


  • The first 7 terms of an arithmetic sequence.

  • The first 7 terms of an increasing geometric sequence.


    Calculate the common difference d and the common ratio r and write down the inserted terms of the sequences.

    
77.

Find the missing term x so that the sequence 2, 4, x, 16… is geometric. Calculate the sum of the first eight terms of the sequence.

    
78.

We have 6 stones arranged in a row. The lightest weighs . Each subsequent one weighs twice as much as the previous one. How much does the heaviest stone weigh and how much do they all weigh together?

    

79.

Calculate x so that:




forms an increasing geometric sequence.

    
80.

Sequence with general term is given. Write the first four terms of this sequence.

    
81.

In one competition, five competitors shared a prize of .

  • Calculate how much each would get if the amounts form an arithmetic sequence with the difference .

  • Calculate how much each would get if the amounts form a geometric sequence with the common ratio .

  • What percentage of the total prize is the amount of ?

    
82.

The figure shows the sequence of points in the plane. Write down the expression for the general term.


    
83.

The first three terms of the sequence are , where .


  • Calculate to make the sequence geometric and write down the terms.


  • Calculate to make the sequence arithmetic and write down the terms.


  • For , calculate the sum of the first ten terms of the corresponding geometric sequence.

    
84.

Given a sequence with a general term .


  • Show computationally that the number 1.97 is not a term of the given sequence.

  • Compute the sequence limit.

  • Which terms are not in the of the number 2 if ? Write down the answer.

    
85.

In the geometric sequence with common ratio 2, the sum of the first twelve terms is equal to 28665. Write the general term of this sequence. How many initial members of the sequence are less than 3829?

    
86.

Find x so that x + 2, x, x - 1 are the first three terms of a geometric sequence. Write down the terms of the sequence.

    
87.

Let the second term of a geometric sequence be and the fifth term . Calculate the first term, the common ratio, and the sum of the first eighteen terms of this sequence.

    
88.

Two geometric sequences are written in the lines below. Fill in the missing links in these sequences in the boxes. Write down the common ratio of these sequences.


  • 27, 9, 3,


  • 2, , 8, 16,

   

89.

Are the following statements correct or incorrect?


  • If 1, 4, 7 are the first three terms of the arithmetic sequence, then the fifth is term 13.


  • The common difference of the arithmetic sequence 2, 0, -2, -4, ... is equal to 2.


  • The first term of the geometric sequence is 4 and the second term is 2. The fourth term of this sequence is 16.


  • The coefficient of the geometric sequence with the first three terms 2, 4, 8 is equal to 2.

    
90.

Let be a geometric sequence with quotient and first term . Let there be a sequence with the general term .


  • Write down the general term of the sequence and prove that the sequence is arithmetic with difference .


  • Calculate the sum of the first 100 terms of the sequence .


  • Prove that for any pair of natural numbers m, n, and are not odd numbers.


  • Let for every natural number n be a polynomial defined by the rule . Prove that for every natural number n the polynomial on the interval has exactly one zero.

    
91.

Solve the following tasks from sequences.


  • Three consecutive terms of an increasing geometric sequence have a sum of 52. If we add 1 to the first term, 8 to the second, and subtract 1 from the third, we get consecutive terms of an arithmetic sequence. Calculate the first three terms of both sequences.


  • The interior angles of the triangle , and are successive terms of an arithmetic sequence. Prove that the sides a, b and c of this triangle have the relation .


  • Calculate all values ​​of the parameter m, so that the solutions of the equation consecutive terms of an arithmetic sequence.