QUADRATIC SEQUENCES: EXAMPLES, RULE AND SOLVED EXERCISES - MATHS - 2025

The Quadratic successions, in mathematical terms, consist of sequences of numbers that follow a certain rule arithmetic. It is interesting to know this rule to determine any of the terms of a sequence.

One way to do this is to determine the difference between two successive terms and see if the value obtained is always repeated. When this is the case, it is said to be a regular sequence.

Number sequences are a way of organizing sequences of numbers. Source: pixabay.com

But if it does not repeat itself, then you can try to examine the difference between the differences and see if this value is constant. If so, then it is a quadratic sequence.

Examples of regular sequences and quadratic sequences

The following examples help clarify what has been explained so far:

Example of regular succession

Let the sequence S = {4, 7, 10, 13, 16, ……}

This sequence, denoted by S, is an infinite number set, in this case of integers.

It can be seen that it is a regular sequence, because each term is obtained by adding 3 to the previous term or element:

4

4 + 3 = 7

7+ 3 = 10

10+ 3 = 13

13+ 3 = 16

In other words: this sequence is regular because the difference between the next term and the previous one gives a fixed value. In the example given this value is 3.

The regular sequences that are obtained by adding a fixed quantity to the previous term are also called arithmetic progressions. And the difference -constant- between successive terms is called the ratio and is denoted as R.

Example of non-regular and quadratic sequence

See now the following sequence:

S = {2, 6, 12, 20, 30,….}

When successive differences are calculated, the following values ​​are obtained:

6-2 = 4

12-6 = 6

20-12 = 8

30-20 = 10

Their differences are not constant, so it can be said that it is a NOT regular sequence.

However, if we consider the set of differences, we have another sequence, which will be denoted as S _diff:

S _dif = {4, 6, 8, 10,….}

This new sequence is indeed a regular sequence, since each term is obtained by adding the fixed value R = 2 to the previous one. That is why we can affirm that S is a quadratic sequence.

General rule for constructing a quadratic sequence

There is a general formula to construct a quadratic sequence:

T _n = A ∙ n ² + B ∙ n + C

In this formula, T _n is the term at position n of the sequence. A, B and C are fixed values, while n varies one by one, that is, 1, 2, 3, 4,…

In the sequence S of the previous example A = 1, B = 1 and C = 0. From there it follows that the formula that generates all the terms is: T _n = n ² + n

That is to say:

T ₁ = 1 ² + 1 = 2

T ₂ = 2 ² + 2 = 6

T ₃ = 3 ² + 3 = 12

T ₅ = 5 ² + 5 = 30

T _n = n ² + n

Difference between two consecutive terms of a quadratic sequence

T _{n + 1} - T _n = -

Developing the expression through remarkable product remains:

T _{n + 1} - T _n = A ∙ n ² + A ∙ 2 ∙ n + A + B ∙ n + B + C - A ∙ n ² - B ∙ n - C

By simplifying it, you get:

T _{n + 1} - T _n = 2 ∙ A ∙ n + A + B

This is the formula that gives the sequence of the differences S _Dif that can be written like this:

Dif _n = A ∙ (2n + 1) + B

Where clearly the next term is 2 ∙ Sometimes the previous one. That is, the ratio of the sequence of differences S _diff is: R = 2 ∙ A.

Solved problems of quadratic sequences

Exercise 1

Let the sequence S = {1, 3, 7, 13, 21, ……}. Determine if:

i) Is it regular or not

ii) Is it quadratic or not

iii) It was quadratic, the sequence of differences and their ratio

Answers

i) Let's calculate the difference between the following and the previous terms:

3-1 = 2

7-3 = 4

13-7 = 6

21-13 = 8

We can affirm that the sequence S is not regular, because the difference between successive terms is not constant.

ii) The sequence of differences is regular, because the difference between its terms is the constant value 2. Therefore, the original sequence S is quadratic.

iii) We have already determined that S is quadratic, the sequence of the differences is:

S _dif = {2, 4, 6, 8,…} and its ratio is R = 2.

Exercise 2

Let the sequence S = {1, 3, 7, 13, 21, ……} from the previous example, where it was verified that it is quadratic. Determine:

i) The formula that determines the general term T _n.

ii) Check the third and fifth terms.

iii) The value of the tenth term.

Answers

i) The general formula of T _n is A ∙ n ² + B ∙ n + C. Then it remains to know the values ​​of A, B and C.

The sequence of differences has ratio 2. Furthermore, for any quadratic sequence the ratio R is 2 ∙ A as shown in the previous sections.

R = 2 ∙ A = 2 which leads us to conclude that A = 1.

The first term of the sequence of differences S _Dif is 2 and must satisfy A ∙ (2n + 1) + B, with n = 1 and A = 1, that is:

2 = 1 ∙ (2 ∙ 1 + 1) + B

solving for B we obtain: B = -1

Then the first term of S (n = 1) is worth 1, that is: 1 = A ∙ 1 ² + B ∙ 1 + C. As we already know that A = 1 and B = -1, substituting we have:

1 = 1 ∙ 1 ² + (-1) ∙ 1 + C

Solving for C we obtain its value: C = 1.

In summary:

A = 1, B = -1 and C = 1

Then the nth term will be T _n = n ² - n + 1

ii) The third term T ₃ = 3 ² - 3 + 1 = 7 and it is verified. The fifth T ₅ = 5 ² - 5 + 1 = 21 which is also verified.

iii) The tenth term will be T ₁₀ = 10 ² - 10 + 1 = 91.

Exercise 3

Sequence of areas for Exercise 3. Source: own elaboration.

The figure shows a sequence of five figures. The lattice represents the unit of length.

i) Determine the sequence for the area of ​​the figures.

ii) Show that it is a quadratic sequence.

iii) Find the area of ​​Figure # 10 (not shown).

Answers

i) The sequence S corresponding to the area of ​​the sequence of figures is:

S = {0, 2, 6, 12, 20,….. }

ii) The sequence corresponding to the consecutive differences of the terms of S is:

S _diff = {2, 4, 6, 8,….. }

Since the difference between consecutive terms is not constant, then S is not a regular sequence. It remains to know if it is quadratic, for which we again do the sequence of the differences, obtaining:

{2, 2, 2, …….}

Since all the terms of the sequence repeat, it is confirmed that S is a quadratic sequence.

iii) The sequence S _dif is regular and its ratio R is 2. Using the equation shown above R = 2 ∙ A, it remains:

2 = 2 ∙ A, which implies that A = 1.

The second term of the sequence of differences S _Dif is 4 and the nth term of S _Dif is

A ∙ (2n + 1) + B.

The second term has n = 2. In addition, it has already been determined that A = 1, so using the previous equation and substituting we have:

4 = 1 ∙ (2 ∙ 2 + 1) + B

Solving for B, we obtain: B = -1.

It is known that the second term of S is worth 2, and that it must fulfill the formula of the general term with n = 2:

T _n = A ∙ n ² + B ∙ n + C; n = 2; A = 1; B = -1; T ₂ = 2

That is to say

2 = 1 ∙ 2 ² - 1 ∙ 2 + C

It is concluded that C = 0, that is to say that the formula that gives the general term of the sequence S is:

T _n = 1 ∙ n ² - 1 ∙ n +0 = n ² - n

Now the fifth term is verified:

T ₅ = 5 ² - 5 = 20

iii) Figure # 10, which has not been drawn here, will have the area corresponding to the tenth term of the sequence S:

T ₁₀ = 10 ² - 10 = 90

References

1. https://www.geogebra.org