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Arithmetic sequences

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Arithmetic sequence

1. General term of an arithmetic sequence

1.1. Common difference of the Arithmetic sequence

1.1.1. Increasing / decreasing arithmetic sequence

2. Arithmetic mean

3. Linear interpolation

4. Forum

View the theory »

Geometric sequence

1. General term of a geometric sequence

1.1. Common ratio of a geometric sequence

2. Geometric mean

2.1. Relationship between geometric and arithmetic mean

2.1.1. Numerical Proof

2.1.2. Graphical proof

3. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:arithmetic_sequences/5320/

Example 1.

Find the fifth, ninth, and general term of the arithmetic sequence
Are the numbers and of the term of a given sequence?

Example 2.

Between the numbers and , insert eight numbers to form an arithmetic sequence. Write down the common difference and the terms of the sequence.

Write the first six terms of the arithmetic sequence with the first term and common difference as respectively.

For given data for arithmetic sequences, find the unknown quantities:

Write the first six terms of the arithmetic sequence with the data .

For the given data of the arithmetic sequences, find the unknown quantities:

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

For the given data of the arithmetic sequences, find the unknown quantities:

Write the first six terms of the arithmetic sequence with the data .

10.

Find the given data of the arithmetic sequences, find the unknown quantities:

11.

In the arithmetic sequence, the fourth term is , and the common difference is . Find the thirteenth term of the sequence.

12.

Write the first six terms of the sequences given by the general term .

13.

For the given sequence of find the general term.

14.

For the given sequence find the general term.

15.

Write down the properties of the sequence given by the general term .

16.

Find the fourth, eighth and general term of the arithmetic sequence

17.

Calculate the arithmetic mean of the numbers and .

18.

Calculate the arithmetic mean of the numbers and .

19.

Find the general term of the arithmetic sequence and write down its properties.

20.

Write a rule for the recursive sequence

21.

Find the general term of the arithmetic sequence and write down its properties.

22.

The graph of an arithmetic sequence is drawn in the figure. Write down the general term of the sequence. How many terms of the sequence are less than ?

23.

Write down the general term of the sequence and show that the th term is equal to .

24.

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

25.

Solutions of the equation are the first three terms of the descending arithmetic sequence. Solve the equation and write the general term of the sequence.

26.

Write the general term of the arithmetic sequence given by:

27.

Write the general term of the arithmetic sequence given by:

28.

From which term onwards are the terms of the sequence greater than ?

29.

Write an arithmetic sequence whose sum of the first three terms is and the sum of the squares of the same terms is .

30.

Between the numbers and , insert numbers to form an arithmetic sequence. The terms of the sequence are thirteen. Write down the common difference of the sequence.

31.

The interior angles of a triangle form an ascending arithmetic sequence with the first term . How much do the other two interior angles of a triangle measure?

32.

How much do the sides of a right triangle with area measure if they form an arithmetic sequence?

33.

In the triangle, the sides form an arithmetic sequence. The largest angle measures , and the length of the middle side is . Find the lengths of the other two sides of the triangle.

34.

The first four terms of the arithmetic sequence are given: .

Write down the next two terms of this sequence and the twentieth term.
What is the sum of the first thirty terms of this sequence?
From which term onwards are all terms greater than ?

35.

The second term of the arithmetic sequence is , and the sum of the squares of the first three terms is . Calculate the first term and the common difference of the sequence.

36.

The sum of the first three terms of an arithmetic sequence is , and the sum of the first and fifth terms is . Calculate the first term and the common difference of this sequence. What is the sum of the first hundred terms of this sequence?

37.

Write the first ten terms of the arithmetic sequence with the first term and the difference . What percentage of these ten numbers are divisible by and what percentage are prime numbers?

38.

An arithmetic sequence has a common difference . The sum of the second and third terms is . Calculate the first term of this sequence.

39.

The first term of the arithmetic sequence is and the fifth term is . Calculate the common difference and the hundredth term of this sequence.

40.

In the arithmetic sequence , , , , ... calculate , , , and the sum of first elements.

41.

In the following three lines we have written the terms of three arithmetic sequences. Write the missing terms in these sequences in the blank boxes. At the end of each line, write the common difference of the sequence of that line.

42.

Solve the following arithmetic sequences.

Calculate and if and are consecutive terms of an arithmetic sequence.
Calculate if applies.
The sum of the first five terms of an arithmetic sequence is . From such sequences, choose the one that has the sum of the squares of the first three terms least, and write down its first five terms.

43.

Write the missing terms of the final arithmetic sequence, write down its difference and draw a graph of the given sequence.

44.

Andrey had a notebook with 14 lines on each page. He wrote 1 letter in the first line on the first page, and in each subsequent line 3 more letters than in the previous line.

Calculate the number of letters Andrey wrote in the 9th line on the first page.
Calculate the number of letters Andrey wrote on the first page. Could he have written 300 letters on the first page?
Andrey finished writing the letters when he wrote 37 letters in a line. How many lines did Andrey list in this way?

45.

In the arithmetic sequence we know the first three terms:

Calculate the value of the expression .
How many initial terms must we add to make the sum equal to 0?
Which term of the sequence has the value - 16?

46.

In the arithmetic sequence with the difference :

Calculate the first term of the sequence .

47.

Two arithmetic sequences are given below. Write the missing terms in each of these sequences.

48.

The first four terms of an arithmetic sequence are: . Write the general term of this sequence and calculate the sum of the first terms .

49.

An arithmetic sequence has the following terms as and . Calculate the common difference d and the sum of the first four terms of the sequence.

50.

Calculate for which value of x in are consecutive terms of an arithmetic sequence.

51.

Students collected voluntary contributions and 10 students participated. The amounts collected represent an ascending arithmetic sequence. The minimum amount was € 10 and the fifth was € 20.

What is the tenth amount?
How much did they all collect together?
What percentage of the funds raised represents the maximum amount?

52.

The lengths of the sides of the triangle represent the first three terms of the arithmetic sequence. The perimeter of this triangle measures 21 cm, and the shortest side measures 4 cm. Calculate the lengths of the sides of the triangle.

53.

The numbers , x, y are the first three terms of the arithmetic sequence, their sum is 6. Calculate x, y and the fourth term and write the general term of this sequence.

54.

When building a 26 m high factory chimney, the first meter costs 8000 euros, and each subsequent meter 3000 euros more than the previous meter.

How much does the last meter of chimney cost?
How much does it cost to build a chimney in its entirety?
Could the first ten meters of chimney be built for 210000 euros? Justify the answer by calculation.

55.

Solve these problems from arithmetic sequences:

For what real numbers x is the sequence arithmetic?
What is the sum of all natural numbers between 1000 and 10000 divisible by 17 ?
The lengths of the sides of the triangle form an arithmetic sequence, the middle measures 7.5 cm. The area of the triangle is equal to . Accurately calculate the lengths of the shortest and longest sides of the triangle.
Write down the first term , the common difference d and the general term of an arithmetic sequence whose sum of the first n terms is equal to .

56.

The figure shows the sequence of points in the plane. Write down the expression for the nth term.

57.

Solve these tasks from the sequences:

The sides of a right triangle form an arithmetic sequence with a common difference 3. Calculate the sides of this triangle.
Prove that the equation has real solutions if the real numbers a, b, c are consecutive terms of an arithmetic sequence.
How many solutions does the equation have if the real numbers a, b, c are consecutive terms of a geometric sequence
In a sequence of four real numbers, the first three form a geometric sequence and the last three form an arithmetic sequence. The sum of the first and last numbers is 14, and the sum of the middle two is 12. Calculate these four numbers.