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

Arithmetic and geometric progressions

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:arithmetic_and_geometric_progressions/5415/


Example 1.

The sequence is given


  • Find out what type of sequence it is.

  • Calculate the limit: .

    
Example 2.

We have a sequence with the general term:




  • Write the first four terms of the sequence in three places exactly.

  • Calculate the limit of the sequence.

  • Find out from which term the values of the terms of the sequence differ from the limit by more than ?

    
3.

Write down the rule for the recursive sequence

   
4.

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

    
5.

The sum of the first terms of a non-constant arithmetic sequence is equal to . The first, third, and seventh terms form a geometric sequence. Write the general term of the arithmetic sequence.

    
6.

Below is the formula for the nth term of a sequence:




Calculate the first four terms and draw the sequence graph.

    
7.

We have a sequence with the general term .


  • Calculate the limit of the sequence.


  • Find out which members of the sequence lie inside the ?

    
8.

What are all natural numbers less than that, when divided by , give the remainder ? Write a general sequence member.


Check the comments on problem 32. Use the correct terms.    

9.

The following sequence is given:




  • Calculate its limit, if any.

  • How many members of the sequence are outside the - neighborhood limits for ?

    
10.

Insert three numbers between the numbers and so that all the numbers together form a geometric sequence. Write down the common ratio of the sequence.

    
11.

Given a sequence whose general term is:




and its limit equal to , determine which members of the sequence satisfy the condition .

    
12.

In an arithmetic progression, the third term is more than the first. The second, fifth, and fourteenth terms, on the other hand, consist of successive members of a geometric sequence. Write the general terms of both sequences.

    
13.

Show that for the values of the expressions are consecutive terms of an arithmetic sequence. Write down the common difference of the sequence.

    
14.

Insert six numbers between the numbers and to form an eight-termed finite geometric sequence. Write down the common ratio.

    
15.

Jernej built a tower of cubes. He built the first from 5 cubes, the second from 8 cubes, the third from 11 cubes, and so on. Calculate how many dice Jernej needed for the ninth tower, which formed an arithmetic sequence with the previous eight .

    
16.

The zeros of the polynomial are the first three terms of a geometric sequence. Write down the zeros and general terms of the geometric sequence.

    
17.

What are all natural numbers greater than and less than that are powers of ? Write the general term of the sequence.

    
18.

Given the arithmetic sequence -3, -1, 1, 3, 5, ...


  • Write down the first term and difference of the sequence and calculate the tenth term of the sequence.


  • Which term of the sequence is equal to 243?


  • How many initial terms of the sequence must we add so that their sum is 252?

    

19.

Calculate for which real numbers x are the values of the given expressions consecutive terms of the arithmetic sequence:


    
20.

Write the first four terms of a geometric sequences for which the sum of the first three terms is and the product of the first and third term is .

    
21.

All terms of an arithmetic sequence are positive numbers. The product of the first and third terms of this sequence is 5, and the sum of the fifth and seventh terms is 10. Calculate the first term and the difference of this sequence. Write down the general term and the three hundredth term of this sequence.

    
22.

The values of the expressions are consecutive terms of a geometric sequence. Write down the terms of the sequence and calculate the fourth term.

    
23.

Calculate the geometric mean of the numbers and .

    
24.

Janez did squats for the morning workout. He made 6 of them on the first day, and then two more each day than the day before. He increased the number of squats to the number 60, and then did 60 each day.


  • Calculate which day John did 60 squats for the first time.


  • Calculate how many squats John did in the first 40 days.


  • John spent 2 seconds for each squat. Was a minute and a half enough for all the squats on the 20th day? The answer is computationally justified.

    
25.

Calculate for which real numbers of x are the values of the given expressions consecutive terms of an arithmetic sequence:


    
26.

Calculate the geometric mean of the numbers and .

    
27.

Calculate for which real numbers x are the values of the given expressions consecutive terms of the arithmetic sequence:


    
28.

The numbers and make up a finite arithmetic sequence. Calculate x and write down the sequence.

    

29.

Calculate for which real numbers x are the values of the given expressions consecutive terms of the arithmetic sequence:


    
30.

Calculate for which real numbers x are the values of the given expressions are consecutive terms of an arithmetic sequence:


    
31.

How many numbers between 100 and 1000 are divisible by 17? Calculate their sum.

    
32.

Calculate for which real numbers x are the values of the given expressions consecutive terms of an arithmetic sequence:


    
33.

Calculate for which real numbers x are the values of the given expressions consecutive terms of an arithmetic sequence:


    
34.

Calculate to what term are the terms of the sequence greater than ?

    
35.

Solution of the equation is the first term, solution of the equation is the second term of a geometric sequence. Write down the common ratio of the sequence.