mathematics
  Personalized »

KS3
 
 
 
Year 7
 
 
Year 8
 
 
Year 9

KS4
 
 
 
Year 10
 
 
Year 11

A-levels
 
 
 
Year 12 (AS)
 
 
Year 13 (A-levels)


physics
  Personalized »

 
 


KS4
 
 
 
KS4 Level Physics

A-levels
 
 
 
AS Level Physics
 
 
A-levels Physics


 
 
Private tutors
 
 
Find a tutor »


 


Unlock access to content
 
Select one option:



Free of charge
Unlock the chapter   "Infinite Geometric Series"  



Unlock access to the solution of problems in this chapter.
Free of charge Price: 3.00 £

Continue »
Subscribe to OpenProf     If you do not get a better grade, you get your money back!
 


Unlock access to all problems solutions.
Available from 8.25 £ / month.
Continue »
mathematics UK Year 10 Mathematics (KS4)
 
Solved problems for the chapter:
 

Infinite Geometric Series

download pdf
View the theory »
Geometric sequence
open
↓   Show theory [1]   ↓


Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:infinite_geometric_series/3603/


Example 1.

Lojze started saving and saved one euro today, he will save two euros tomorrow, three euros the day after tomorrow... How much will he save in one non-transferable year?

    
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.

    
3.

Evaluate:

   
4.

Evaluate:

   
5.

Evaluate:


    
6.

Evaluate:


    
7.

Evaluate:


    
8.

Evaluate:


    

9.

Evaluate:


    
10.

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


    
11.

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


    
12.

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


    
13.

The sum of the first two terms of a geometric sequence is 14. If the sum to infinity of the sequence is 32, obtain the sequence.

    
14.

Solve:


    
15.

Solve:


    
16.

Express as a fraction.

    
17.

Express as a fraction.

    
18.

Express as a fraction.

    

19.

For what value of x is equal to 24.

    
20.

For what value of x will the series be convergent. Hence, calculate the sum.

    
21.

Given:




  • At what value x will the series converge?


  • Calculate the sum of the geometric sequence.


  • For what value of x the sum equal to 25.

    
22.

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


    
23.

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.


    
24.

Calculate the length of the lines in the figure below.


    
25.

Given the sequence .


  • Find the range of values of x for which the given sequence converges?


  • Calculate the value of x such that the sum of the terms is twice the first term.


  • Calculate the value of x for which the square of the first term is seven times the fifth term. Round the result to four decimal places.


  • For x = -1 obtain the sum of the first n terms of the given sequence?


  • What is the least number of terms that must be added so that the sum will differ from the sum to infinity by less than ?

    
26.

Given the series:




  • Calculate the exact sum of the series when




  • For what value of x is the sum equal to 1?

  • Calculate for what values x the given sequence converges.

    
27.

Given the series:




  • For what real values of x is the series convergent?


  • Calculate the value of for which the sum is .


  • For x = -1, what percentage of the sum of all members of this series is the sum of the first five terms? Round the result to 5 significant figures.


  • For x = -1, how many members are smaller than ?

    
28.

Given the real function .


  • Solve the equation . Give an exact answer.


  • Prove that .


  • Evaluate .


  • Find the range of values of x for which converges, and calculate for which the sum to infinity is 1.

    

29.

A sequence is given by the general term .


  • Obtain the first three terms of the given sequence.


  • Compute the limit of the given sequence.


  • Calculate the sum .

    
30.

Given a geometric sequence whose first term is x and second term is , for all real x.


  • Find all real numbers x for which the second term is 15.


  • Find all real numbers x for which the corresponding geometric series is convergent. Give the exact answer.


  • Let be and let the corresponding geometric sequence have a general term . Find the smallest natural number n for which the difference is less than .