What is a quadratic sequence ? Formula for n-th term and sum of n terms of a quadratic sequence with solved examples

Before we begin with the term Quadratic Progression. Let's first get over to progression and its terminology 

What is Sequence/Progression 

A Sequence is an ordered list of numbers/items that follow a specific rule.The sequence could be an arithmetic progression, a geometric sequence or harmonic sequence.

Of the three sequences (A.P ,G P and H.P) . arithmetic sequence is instrumental in understanding the concept of a quadratic sequence 

Arithmetic progression : A progression/sequence where the difference between two consecutive terms is always a constant called arithmetic progression.

Example : 12,16,20,24...

Observe the difference between two consecutive terms 

16-12=4 , 20-16=4 , 24-20=4 

Hence the sequence is an A.P

Now you know the arithmetic sequence, let's define a quadratic sequence.

What is a Quadratic sequence 

A Sequence is called a quadratic sequence if the difference between two consecutive terms of the sequence always follows an arithmetic progression.

In simple terms, If we obtain an Arithmetic progression from the difference of two consecutive terms of a given sequence then we say the sequence is a Quadratic sequence otherwise not.

Example 1 : consider the following sequence 

3,7,13,21,31,43...............

First difference of two consecutive terms:

7-3 = 4 

13-7 = 6

21-13 = 8 

31-21 = 10

43-31 =12.. 

The series obtained from the first difference is : 

4,6,8,10,12...

Now let's find Second difference of two consecutive terms:

6-4 = 2 

8-6=2

10-8=2

12-10 =2

Since we can see the second difference of two consecutive terms is a constant value(definition of A.P), hence the given sequence is a quadratic sequence.

Example 2 : Look at the following sequence 

1,3,7,13,21,31,43,57........

First difference :

3-1=2

7-3=4

13-7=6

21-13=8

31-21=10

43-31=12

57-43=14

Series form from difference is :

2,4,6,8,10,12,14

Second difference :

4-2 =2

6-4= 2

8-6= 2

10-8= 2

12-10 =2

14-12= 2

the second difference is a constant value hence the given sequence is quadratic sequence 

Formula for n-th term of a quadratic sequence

The n-th term of a quadratic sequence is given by the following expression 

\( A_n \) = \( An^2 + Bn + C  \)

Where A ,B and C are constants ,let's see how the values for these constants are computed.

\( A = {\text{Scecond Difference of a quadratic sequence} \over 2}\) 

\( B \) = The difference between the second term \(a_2\) and the first term \(a_1\) of a quadratic sequence \( -3A \)

\( C \) = First term of a quadratic sequence\(a_1\)  - \((A+B)\) 

Let's take some examples as to how we can find the n-th term of a given quadratic sequence

Example 1: Find 18th term of the given quadratic sequence : 4,11,20,31,44.....

Ans: Sequence - 4, 11,20,31,44.....

First difference of this given sequence is 

11-4 =7 
20-11 =9
31-20 =11
44-31 =13

Second difference 

9-7 =2
11-9 =2 
13-11=2 

Second difference : 2

Formula for the n-th term of a quadratic sequence is : \( A_n \) = \( An^2 + Bn + C  \)

\( A = {\text{second difference} \over 2} = {2 \over 2} =1   \) 

\( B = \text{second term } a_2\)
\(- \text{first term } a_1 - 3A \) 
 = \({11 - 4 } -{3 \times 1 } \) 
= 4

\( C = \text{first term } a_1 - (A+B) \)
= 4 -(4+1) = -1 

therefore the n-th term for the sequence will be given by : 

\( A_n \) = \( n^2 + 4n - C  \)

To find the 18-th term of this sequence we put n =18 in the above equation.

\( A_{18} \) = \( 18^2 + 4 \times 18 +{(-1)} \)

= 364 + 72 -1
= 395

Example 2 : The sequence 25, 20, 13, 4, -7... is a quadratic sequence, Find its 50th term. 

Ans : 

First difference:  -5,-7,-9,-11...

Second difference: -2

A = \( -2 \over 2\) = -1

B = \(a_2-a_1 - 3A \) 

= \( 20-25- 3\times(-1) \)

= -2

C = \(a_1 \) -\( (A+B) \)
= 25 - (-1+(-2)) 
= 25 +3
=28 

therefore, the n-th term for this quadratic sequence is given by the following quadratic equation : \( -n^2 -2n +28 \) 

To find the 50-th term of this sequence we put n =50.

\( A_{50} \) = \( -(50)^2 -2 \times 50 +28  \)

= -2600-100 +28 
= -2572

Sum of n-terms of a quadratic sequence

we have learnt the formula for a quadratic sequence now it is time to drive a formula for the sum of specified terms of that sequence. By specified terms we mean that it may vary-for example ,sum of 2 terms or  sum of a large number of terms.

k-th term of a quadratic sequence : 

\( A_k \) = \( Ak^2 + Bk + C \)

the sum of n-terms of a quadratic sequence \( S_n \) :

\( \sum A_{k=1}^{n}  = A \sum_{k=1}^{n} k^2 +
B \sum_{k=1}^{n} k + \)

\(C \sum_{k=1}^{n} 1 \)

\( \sum_{k=1}^{n}k = { n(n+1) \over 2} \)  (Sum of n natural numbers ) 

\( \sum_{k=1}^{n} k^2  = { n(n+1) (2n+1) \over 6 }\) (Sum of the squares of n natural number)

\( \sum_{k=1}^{n} 1 = n \) ( number 1 repeating n times )

\( S_n = A \left[ \frac{n(n+1)(2n+1)}{6} \right]
+ B \left[ \frac{n(n+1)}{2} \right] \)

\(+ Cn \)

This is the expression through which we can find the sum of n-terms of a quadratic sequence.

Here are some solved examples based on the sum of n-terms of a quadratic sequence.

Example 1: The expression \( 2n^2 - 2n + 5 \) represents a quadratic sequence. Find the sum of the first 10 terms of this sequence.

Ans: Let's compare the expression with the n-th term of a quadratic sequence.

Compare  \( 2n^2 - 2n + 5 \) with  \( An^2 + Bn + C \) 
we get 
A = 2 , B =-2  , C = 5 

We know the  formula for  the sum of n terms of a quadratic sequence.

\( S_n = A \left[ \frac{n(n+1)(2n+1)}{6} \right] +
B \left[ \frac{n(n+1)}{2} \right] \)
\( + Cn \)

Sum of first 10 terms of a quadratic sequence is 

\( S_{10} = 2 \left[ \frac{10(10+1)(2 \times 10+1)}{6} \right] -2 \left[ \frac{10(10+1)}{2} \right] \)
\( + 5 \times 10 \)

= \( 2 \left[ \frac{10(11)(21)}{6} \right] \) \( -2 \left[ \frac{10(11)}{2} \right] \) +50
= 770 -110 + 50
= 660+50 
= 710

The sum of the first 10 terms of the given quadratic sequence is 710.

Example 2: Determine the sum of the first 40 terms of the quadratic sequence: 6,19,40,69...

Ans:

First difference:  13,21,29...

Second difference: 8

Let's find the quadratic equation for the n-th term of this quadratic sequence

n-th term of the quadratic sequence : \( A_n \) = \( An^2 + Bn + C  \)

A = \( 8 \over 2\) = 4

B = \(a_2-a_1 - 3A \) 

= \( 19-6- 3\times 4 \)

= 1

C = \(a_1 \) -\( (A+B) \)

= 6 - (4+1) 

= 1

quadratic equation for the n-th term of the given sequence is : \( 4n^2 + n + 1  \)

now we calculate the sum of first 40 terms of this sequence: 

\( S_n = A \left[ \frac{n(n+1)(2n+1)}{6} \right] + B \left[ \frac{n(n+1)}{2} \right] + \)
\( Cn \)

n = 40 , A = 4, B = 1 and C =1 

\( S_{40} = 4 \left[ \frac{40(40+1)(2 \times 40+1)}{6} \right]
+ 1 \left[ \frac{40(40+1)}{2} \right] \)
\( + 1 \times 40 \)

= \( 4 \left[ \frac{40(41)(81)}{6} \right] \) + \( \left[ \frac{40(41)}{2} \right] \)
+ 40
= 88560 + 820+40 
= 89420

Frequently Asked questions on Quadratic Sequence

Q1: How can you tell if a sequence is quadratic?

Ans: A sequence is quadratic if its second differences are constant.

Q2 : What is the general formula for a quadratic sequence?

Ans: \( A_n \) = \( An^2 + Bn + C  \)

Q3 : How to find the next term in a quadratic sequence?

Ans: To find the next term of a quadratic sequence ,let's take an example

Consider this Sequence: 2, 7, 16, 29

First differences: 5, 9, 13

Second differences: 4, 4

Next first difference = 13 + 4 = 17

Next term = 29 + 17 = 46

Q4 : Can a quadratic sequence contain negative numbers?

Ans: Yes. A quadratic sequence can contain negative numbers 

Example : -9, -6, -1, 6, 15, ...

Q5 : What is the difference between an arithmetic sequence and a quadratic sequence?

Ans:  An arithmetic sequence has a constant first difference between terms, while a quadratic sequence has a constant second difference.

Q6 : Can a quadratic sequence decrease before increasing?

Ans:  Yes. Some quadratic sequences decrease at first and then increase

Example : Parabola with a minimum point (vertex)
Sequence : 5, 2, 1, 2, 5, 10,...