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

Sum of terms of a progression

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Mathematical induction
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Arithmetic sequence
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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:sum_of_terms_of_a_progression/5416/


Example 1.

Calculate:


    
Example 2.

Calculate .

    
3.

Calculate .

    
4.

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?

    
5.

Calculate .

    
6.

Numbers 2, 5, 8, 11 are the first four terms of an infinite arithmetic sequence.

  • Write the general term of this sequence.


  • Calculate which term of this sequence is equal to 6041.


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

    
7.

With mathematical induction, show that:


    
8.

Add all natural numbers less than 100 that are divisible by 6.

    

9.

With mathematical induction, show that:


    
10.

Using mathematical induction, show that:


    
11.

Evaluate:


    
12.

With mathematical induction, prove that the sum of the first n terms of the geometric sequence is equal to:


    
13.

Prove by induction that the sum of the first n terms of the arithmetic sequence is equal to:


    
14.

Calculate , find the formula for the sum of the given finite terms and prove that the formula is general by mathematical induction:


    
15.

Evaluate:


    
16.

Calculate the sum of the terms of if and

   
17.

Calculate , find the formula for the sum of the given finite terms and prove the correctness of the formula by mathematical induction:


    
18.

Given an arithmetic sequence :


  • Calculate the thirteenth term and

  • the sum of the first thirteen terms of this sequence.

  • How many terms of this sequence are less than 1000?

    

19.

Calculate . Find the formula for the sum of the given series and prove the correctness and generality of the formula by complete induction:


    
20.

Calculate . Find the formula for the sum of the given finite series and prove the correctness and generality of the formula by mathematical induction:


    
21.

The third term of an arithmetic sequence is equal to 8, and the fifth term is equal to 15. Calculate the common difference, the first term and the sum of the first 100 terms of the given sequence.

    
22.

With mathematical induction prove:


, for each

   
23.

Solve the equation .

    
24.

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.

    
25.

Prove:


  • that the sum of four powers of 5, whose exponents are consecutive natural numbers, is divisible by 26.


  • by mathematical induction that for every natural number n.

    
26.

Calculate the sum of the first 50 natural numbers that give a remainder 2 when divided by 5.

    
27.

Calculate:


  • The sum of the first 100 natural numbers.


  • The sum: .

    
28.

Solve:


    

29.

In the figure, the ordinates of the columns consist of a finitely descending arithmetic sequence. Write down the general term of the sequence. How many members does the sequence have? What is the sum of the areas of the columns?


    
30.

Calculate

   
31.

Insert some numbers between numbers 5 and 44 to form a finite arithmetic sequence. What is the common difference of the sequence and how many terms are inserted if the sum of the inserted terms is 294.

    
32.

Solve:


    
33.

Between the numbers 13 and 97 we insert n numbers to form the finite arithmetic sequence. Calculate the arithmetic sequence difference if the sum of the inserted numbers is 1100.

    
34.

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

    
35.

Solutions of the equation:




are the first three terms of the ascending arithmetic sequence. Solve the equation and write down the general term of the sequence. What is the sum of the first 10 members of the sequence?

    
36.

For what value of x is equal to 24.

    
37.

Calculate . Find the general formula for the sum of the given finite terms and prove the correctness of the formula by induction:


    
38.

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

    

39.

How many terms of the arithmetic sequence 2, 10, 18, 26, 34… do we need to add to make their sum greater than 380?

    
40.

Solution of the equation:




are the first and fourth terms of the descending arithmetic sequence. Solve the equation and write down the general term. What is the sum of the first 20 members of the sequence?

    
41.

Show that the sequence with the sum of is arithmetic. Write down the first term and the difference .

    
42.

Calculate:


    
43.

Calculate:


    
44.

Calculate:


    
45.

In arithmetic order, the first term is 4 and the fifth is 24. What is ?

    
46.

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.

    
47.

How many terms of an arithmetic sequence with sum -16 must be added if and ?

    
48.

The first term of the arithmetic sequence is 43, the number of terms is 25, and the sum is 175. Calculate the difference and the last term of the sequence .

    

49.

What is the first and how much is the last term of a 31-term arithmetic sequence with sum 2542 and difference 5?

    
50.

The last term of the final arithmetic sequence with the sum of 167500 and the difference is 334. Find the first term and the number of terms .

    
51.

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 .