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mathematics UK Year 12 Mathematics (AS)
 
Solved problems for the chapter:
 

Geometric Series

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Geometric sequence
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Infinite geometric series
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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:geometric_series/4749/


Example 1.

Calculate:


    
Example 2.

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?

    
3.

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.

    
4.

Evaluate:

   
5.

Evaluate:

   
6.

Evaluate:


    
7.

Evaluate:


    
8.

Evaluate:


    

9.

Evaluate:


    
10.

Evaluate:


    
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 sequence with the following data:


    
13.

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


    
14.

The sum of the first two terms of a geometric sequence is 78. If the sum to infinity of the sequence is 294, obtain the sequence.

    
15.

We have the first two terms of a sequence as and .


  • If and are the first two terms of the arithmetic sequence. Write the fifth term of this sequence and calculate the sum of the first hundred terms.


  • If and are the first two terms of the geometric sequence. Write down the fourth term of this sequence and calculate the sum of the first fifteen terms.

    
16.

Solve:


    
17.

Calculate

   
18.

Solve:


    

19.

In ascending geometric order, the first term is and the fifth is . How much is ?

    
20.

For what value of x is equal to 24.

    
21.

How many terms of the geometric sequence with the sum of must we add if and ?

    
22.

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

    
23.

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 40.

    
24.

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


    
25.

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.


    
26.

Calculate the length of the lines in the figure below.


    
27.

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 ?

    
28.

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.

    

29.

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 ?

    
30.

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.

    
31.

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 .

    
32.

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 .