Linear, Quadratic and Cubic sequences
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This document defines and provides examples of different types of numeric sequences: - Linear/arithmetic sequences have a constant difference between consecutive terms. Formulas use the first term and common difference. - Quadratic sequences have a constant second difference; their formulas are of the form an^2 + bn + c. - Cubic sequences have a constant third difference; their formulas are of the form an^3 + bn^2 + cn + d. Step-by-step methods are provided for determining the formulas for quadratic and cubic sequences based on analyzing differences between terms. Worked examples demonstrate finding common differences and deriving sequence formulas.
![Cubic sequences:
Sequences where the 3rd difference is constant are known as cubic sequences.
Cubic sequences are of the form:
an3
+ bn2
+ cn + d
Examples:
0 12 10 0 -12 -20
1st difference +12 -2 -10 -12
2nd difference -14 -8 -2
3rd difference +6 +6
3rd difference is constant so it is a cubic sequence .
It is of the form: an3
+ bn2
+ cn + d = n3
-13n2
Step 1: Value of a= 1/6 X( 3rd difference) = 1/6 X (6) = 1 . Hence a=1.
Step 2: Write original sequence and subtract an3
i.e. n3
from it as shown below.
S 0 12 10 0 -12 -20
1n3
1 8 27 64 125 216
s-1n3
= our new sequence -1 +4 -17 -64 -137 -236
1st Common difference
2nd common difference
+5 -21 -47 -73
-26 -26 -26
2nd common difference 2nd difference is constant so our new sequence is quadratic and of the
form bn2
+ cn +d; where b=-26/2 = -13
Performing this operation:
[S-( n3
-13n2
)];
leaves us with a linear
sequence and we get the
part cn+d
S 0 12 10 0 -12
n 1 2 3 4 5
( n3
-13n2
) (1-13)=
-12
8-52=
-44
27-117=
-90
64-208
=-144
125-325
=200
Finally the required
sequence is
s-( n3
-13n2
) 12 56 100 144 -212
Common difference is
44.Sequence is linear +44 +44 +44 +44
Formula for the linear
sequence is 12 + (n-1) 44 = 12 + 44n-44 = 44n+32
Ans: Cubic sequence is n3
- 13 n2
+ 44n +32
w
w
w
.sm
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Linear, Quadratic and Cubic sequences
- 1. SEQUENCES The following worksheets contain solved examples for linear , quadratic and cubic sequences. w w w .sm arteduhub.com www.smarteduhub.com 1
- 2. Sequences Sequence: A sequence is a list of numbers (or other things) that changes according to some sort of pattern In mathematics this pattern is called a common difference or ratio. Term: Each number in a sequence is called as a term. For example the first term is the term that occurs first in a sequence. The 5th term is the term that occurs in the fifth place of the sequence. The nth term is the term that occurs in the nth position of the sequence. Examples of sequences: 1,4,9,16,25,36......... 0,2,4,6,8,10........ Types of sequences: Arithmetic (Linear) Quadratic Cubic Geometric w w w .sm arteduhub.com www.smarteduhub.com 2
- 3. Linear sequence: A linear sequence is a sequence with the first difference between two consecutive terms constant. Examples: a. 0,2,4,6,8,10,....... 0, (0+2), (2+2), (4+2)..............the common difference is 2 b. 30,35,40,45,50,55,....... 30, ( 30+5), (35+5), (40+5),......the common difference is 5 c. 48, 46, 44,42,....... 48, (48-2), (46-2), (44-2),.......the common difference is -2 Formula for calculating the nth term of an arithmetic ( linear sequence) is a+( n-1) d where: a= 1st term, d= common difference & n = the term you need to find out. Example: In the sequence 'a' above; first term is =a=0, d= common difference =2 and n=7th term would be calculated in the following way: tn= a+(n-1)d So, t7= 0+ (7-1)2= 0 + 6(2) = 0+12 =12 Do itself yourself Find the common difference and also the next two terms for the following sequences: −15, −7, 1, 9, ... −40, −240, −440, −640, ... 31, 21, 11, 1, ... 38, 138, 238, 338, ... w w w .sm arteduhub.com www.smarteduhub.com 3
- 4. Quadratic sequence: Whenever the second difference is constant in a sequence, the sequence is said to be a quadratic sequence. In such a sequence the 2nd difference is constant. It is of the form: an2 + bn + c Examples of quadratic sequences: a. 1, 4, 9, 16, 25 ............ +3 +5 +7 +9 -------------> 1st difference is not constant +2 +2 +2 +2 ------------> 2nd difference is constant Method for finding the formula for the sequence: Sequence is of the form: an2 + bn +c 1n2 +1 Step 1 2nd difference is 2. So to find the value of a , always halve the common difference. So value of a is 2/2=1 a=1 Step 2 Write the original sequence and from it subtract the value of 1n2 Original seq: (S) 1 4 9 16 25 n 1 2 3 4 5 n2 1 4 9 16 25 S-(n2 ) 0 0 0 0 0 Step 3 Observe the common difference( d) when you subtract 1n2 from the original sequence Common difference ( d) 0 Step 4 The remaining sequence is therefore linear and of the form a+ ( n-1) d Remaining sequence is : 1+ ( n-1) 0 = 1+ 0= 1 Step 5 Combining steps 1 and 4 we get our required quadratic sequence n2 +1 w w w .sm arteduhub.com www.smarteduhub.com 4
- 5. Cubic sequences: Sequences where the 3rd difference is constant are known as cubic sequences. Cubic sequences are of the form: an3 + bn2 + cn + d Examples: 0 12 10 0 -12 -20 1st difference +12 -2 -10 -12 2nd difference -14 -8 -2 3rd difference +6 +6 3rd difference is constant so it is a cubic sequence . It is of the form: an3 + bn2 + cn + d = n3 -13n2 Step 1: Value of a= 1/6 X( 3rd difference) = 1/6 X (6) = 1 . Hence a=1. Step 2: Write original sequence and subtract an3 i.e. n3 from it as shown below. S 0 12 10 0 -12 -20 1n3 1 8 27 64 125 216 s-1n3 = our new sequence -1 +4 -17 -64 -137 -236 1st Common difference 2nd common difference +5 -21 -47 -73 -26 -26 -26 2nd common difference 2nd difference is constant so our new sequence is quadratic and of the form bn2 + cn +d; where b=-26/2 = -13 Performing this operation: [S-( n3 -13n2 )]; leaves us with a linear sequence and we get the part cn+d S 0 12 10 0 -12 n 1 2 3 4 5 ( n3 -13n2 ) (1-13)= -12 8-52= -44 27-117= -90 64-208 =-144 125-325 =200 Finally the required sequence is s-( n3 -13n2 ) 12 56 100 144 -212 Common difference is 44.Sequence is linear +44 +44 +44 +44 Formula for the linear sequence is 12 + (n-1) 44 = 12 + 44n-44 = 44n+32 Ans: Cubic sequence is n3 - 13 n2 + 44n +32 w w w .sm arteduhub.com www.smarteduhub.com 5