Finding the Next term in a third difference Sequence (serajian Asl)
A general formula for determining the next terms in a third
difference sequences
In this article we are going to introduce a general formula for determining
the terms in third difference sequences without using of matrix operations
Below is a sample of third difference sequence
45.14
,
74.44
29.3
,
119.2
44.76
15.46
,
179.74 ,
60.54
15.78
0.32
256.38 ,
76.64
16.1
349.44 ,
93.06
16.42
0.32
0.32
126.86
109.8
0.32
730.34 ,
144.24
17.06
16.74
0.32
586.1 ,
459.24 ,
17.38
0.32
892.28 ,
161.94
17.7
0.32
For determining the next term in above kinds of sequences we should divide the
sequence to three separated sequences a , b , c
a1
c1
b1
45.14
74.44
29.3
a2
119.2
44.76
15.46
179.74
60.54
15.78
0.32
c2
b2
256.38
76.64
16.1
16.42
0.32
0.32
a3
349.44
93.06
586.1
126.86
0.32
17.38
0.32
a4
730.34
144.24
17.06
16.74
c3
b3
459.24
109.8
0.32
as below fig
892.28
161.94
17.7
0.32
18.02
0.32
In this case we will have three separated sequences as shown below
a1
45.14
b1
a2
a3
a4
a5
a6
a7
a8
a9
a10
, 179.74 , 459.24 , 892.28 , 1487.5 , 2253.54 , 3199.04 , 4332.64 , 5662.98 , 7198.7 ,
b2
74.44 , 256.38 ,
c1
c2
119.2
, 349.44 ,
b3
b4
b5
b6
b7
586.1 , 1072.24 , 1723.44 , 2548.34 , 3555.58 ,
c3
c4
c5
730.34 , 1270.54 , 1978.68 ,
c6
c7
b8
b9
b10
4753.8 , 6151.64 , 7757.74 ,
c8
c9
c10
2863.4 , 3933.34 , 5197.14 , 6663.44 , 8340.88 ,
a , sequence we will use of the below formula.
for determining the terms in
an


2


2

3

2
a1  9m  51m  72 Xa  9m  45m  56 Ya  ( 3m  7)Za  ( 3m  8)Ta  36m  28 8m  77 0m  68 8 d
And the unknown coefficients values {Xa,Ya,Za,Ta}in formula can be obtain via
down formulas
Xa
an
 c  2b  a 
1
1
 1
d

2
Ya
 a2  c1  b1  a1  2d
2
= the number of nth term in (a,) sequence
m = (n+4)/2

2
Ta
 c1  b1
= the first term in (a,) sequence
d = common difference
For determining the terms in
bn
a1
 b1  a1
Za


2
b , sequence we will use the below formula.


3

2
b1  9m  51m  72 Xb  9m  45m  56 Yb  ( 3m  7)Zb  ( 3m  8)Tb  36m  28 8m  77 0m  68 8 d
And the unknown coefficiens values {Xb,Yb,Zb,Tb} in it can be obtain via down
formulas
Xb
bn
 a  2c  b  d 
1
1
2
 2
Yb
 b2  a2  c1  b1  2d
2
= the number of nth term in (b,) sequence
m = (n+4)/2
For determining the terms in
b1
Zb
 c1  b1
Tb
 a2  c1
= the first term in (b,) sequence
d = common difference
c , sequence
we will use the below formula.
And the unknown cofficients values {Xc,Yc,Zc,Tc} on it can be obtain via down
formulas
cn
= the number of nth term in (c,) sequence
c1
= the first term in (c,) sequence
m = (n+4)/2
d = common difference
Example: Find the terms (16th & 19th & 22th) in below third difference sequence.
a1
45.14
c1
b1
,
74.44
29.3
,
a2
119.2
44.76
15.46
,
179.74 ,
60.54
256.38 ,
76.64
15.78
16.42
0.32
a3
349.44 ,
93.06
16.1
0.32
c2
b2
586.1 ,
459.24 ,
126.86
109.8
17.06
16.74
0.32
0.32
c3
b3
0.32
730.34 ,
144.24
…
892.28 ,
161.94
17.38
0.32
a4
17.7
0.32
a , sequence
As we know the above mentioned terms, (16th & 19th & 22th) are in
a , sequence is (1+3k) and also for
the b , sequence the general fig is (2+3k) and for c , sequence the general
Because the general fig of terms number in
fig is (3+3k)
16th term in main ( a , b , c ) sequence = 6th term in ( a , ) sequence
17th term in main ( a , b , c ) sequence = 6th term in ( b , ) sequence
18th term in main ( a , b , c ) sequence = 6th term in (
c ,)
sequence
Now for this Example: (16 = 1+3*5) and (19 = 1+3*6) and (22 = 1+3*7), it means
that three of 16th & 19th & 22th terms belongs to
of
a , sequence and we should use
a , sequence formulas for determine the values of terms.
an
n
Xa
Ya


2


2

3

2
a1  9m  51m  72 Xa  9m  45m  56 Ya  ( 3m  7)Za  ( 3m  8)Ta  36m  28 8m  77 0m  68 8 d
6 7 8
m
n 4
2
5 5.56
d
 119.2  274.44  45.14 

0.3 2
0.32
2

15.62
179.74  119.2  74.44  45.14  20.32
2
a1
15.3
45.14
a6
a7
a8
Za
( 74.44  45.14)
29 .3
Ta
( 119.2  74.44)
44 .76
a6
45.14   9  52  51  5  72 15 .62  9  52  45  5  56 15 .3 ( 3  5  7) 29 .3 ( 3  5  8) 44 .76  36  53  28 8 52  77 0 5  68 8 0.3 2
a7
45.14   9  5.52  51  5.5  72 15.62  9  5.52  45  5.5  56 15.3 ( 3  5.5  7)29.3 ( 3  5.5  8) 44.76  36  5.53  288 5.52  770 5.5  688 0.32
a8
45.14   9  6  51  6  72 15 .62  9  6  45  6  56 15 .3 ( 3  6  7) 29 .3 ( 3  6  8) 44 .76  36  6  28 8 6  77 0 6  68 8 0.3 2
2
2
3
2
22 53.54
43 32.64
3199.04
a1
45.14
a2
a3
a4
a5
a6
a7
a8
a9
a10
, 179.74 , 459.24 , 892.28 , 1487.5 , 2253.54 , 3199.04 , 4332.64 , 5662.98 , 7198.7 ,
As this procedure we can determine the other terms in sequences a , b , c
For the other samples about third or fourth or more difference sequences
please refer to “another look to Stirling numbers... Stirling cube array” in
present web.sit
modiranhesab
Serajian Asl
Finding the Next term in a third difference Sequence
“Arithmetic sequences” “finding the next term in a third difference sequence”