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

Geometric and Arithmetric Series

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Arithmetic sequence
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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:geometric_and_arithmetric_series/3601/


Example 1.

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

    
Example 2.

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.    
3.

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.

    
4.

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

    
5.

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 .

    
6.

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?

    
7.

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.

    
8.

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.

    

9.

Solve the following tasks:


  • Given a with the parameters , , as measures . Calculate both possible page lengths . Round the result to two decimal places.


  • The lengths of the sides of a triangle are three consecutive terms of an arithmetic sequence with difference d = 1. Calculate the lengths of the sides if the largest angle of the triangle measures .


  • In the triangle measure the sides and . Between the angles and the connection applies. Calculate the area of triangle . Round the result to two decimal places.

    
10.

Solve the following problems in geometry:


  • In the circle with radius we draw a rectangle whose sides are in the ratio . Calculate the perimeter of the rectangle. What percentage of the area of the circle is the area of the rectangle?


  • In a rhombus, the difference of diagonals is , and its area measures . Calculate the lengths of the diagonals and its side. Leave the result in surd form.


  • The sides of the triangle represent the first three terms of an arithmetic sequence with a difference of 4. The plane of the triangle is . Calculate the lengths of the sides.

    
11.

Calculate the sum of the terms of if and

   
12.

The base face of an upright pyramid with volume is a rectangle with sides of lengths a and b. The length of the height of this pyramid is denoted by v. The numbers a, b and v in this order form an increasing arithmetic sequence with a common difference . Calculate the lengths of the base edges and the height of the pyramid.

    
13.

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?

    
14.

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

    
15.

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.

    
16.

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

    
17.

Solve the equation .

    
18.

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

    

19.

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?


    
20.

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.

    
21.

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.

    
22.

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