Mathematics Dictionary
Dr. K. G. Shih
Amicable Numbers
Symbol Defintion
Example : Sqr(x) = square root of x
Q01 |
- Difference of 1, 4, 9, 16, ...
Q02 |
- Difference of 1, 3, 6, 10, 15, ....
Q03 |
- Difference of 1, 8, 27, 64, ....
Q04 |
- References
Q01. Find 2nd difference of sequence 1, 4, 9, 16, 25, 36,...
First diiference and second difference
Position : 01 02 03 04 05 06 .....
Sequence : 01 04 09 16 25 36 .....
1st diff : .. 03 05 07 09 11 .....
2nd diff : ..... 02 02 02 02 .....
Study
This is a quadratic expression y = a*x^2 + b*x + c
We know a = 2/2 = 1 from the 2nd difference
If x = 1 and y = 1
Hence 1 = a*(1)^2 + b*1 + c
Since a = 1
Hence b + c = 0 ........... (1)
If x = 2 and y = 4
Hence 4 = a*(2)^2 + b*(2) + c
Since a = 1
Hence 4 = (1)*(4) + 2*b + c
hence 2*b + c = 0 ......... (2)
Solve (1) and (2)
(2) - (1) : b = 0 ......... (3)
From (1) and (3) we hace c = 0
Hence expressions is
y = a*x^2 + b*x + c
y = (x^2)
This is the nth term of the sequenc T(n) = n*(n+1)/2
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Q02. Find second difference 1, 3, 6, 10, 15, .....
First diiference and second difference
position : 01 02 03 04 05 06 ..... (x value)
Sequence : 01 03 06 10 15 21 ..... (y value)
1st diff : .. 02 03 04 05 06 .....
2nd diff : ..... 01 01 01 01 ..... (It is common difference)
Study
This is a quadratic expression y = a*x^2 + b*x + c
The second derivative is y" = 2
Hence a = 1/2 from the 2nd difference
If x = 1 and y = 1
Hence 1 = a*(1)^2 + b*1 + c
Since a = 1/2
Hence b + c = 1/2 ......... (1)
If x = 2 and y = 3
Hence 3 = a*(2)^2 + b*(2) + c
Since a = 1/2
Hence 3 = (1/2)*(4) + 2*b + c
hence 2*b + c = 1 ......... (2)
Solve (1) and (2)
(2) - (1) : b = 1/2 ....... (3)
From (1) and (3) we hace c = 0
Hence expressions is
y = a*x^2 + b*x + c
y = (x^2)/2 + *x/2
y = x*(x + 1)/2
This is the nth term of the sequenc T(n) = n*(n+1)/2
Go to Begin
Q03. Find the 3rd difference of 1, 8, 27, 64, ....
Fund difference
Position : 01 02 03 04 005 .....
Sequence : 01 08 27 64 125 .....
1st diff : 00 07 19 37 061 .....
2nd diff : 00 00 12 18 024 .....
3rd diff : 00 00 00 06 006 .....
Expression of nth term : T(n) = n^3
y = a*x^3 + b*x^2 + c*x + d
Hence a = 1/3rd difference = 1/6
We can prove that b = 0, c = 0 and d = 0
Hence y = x^3 is expression
The 3rd derivative of y = x^3 is 6
Go to Begin
Q4. References
Difference in linear function sequences : y = a*x + b
* ..... x ....... y ....... y'
* ..... 1 ..... a+b ....... a
* ..... 2 .... 2a+b ....... a
* ..... 3 .... 3a+b ....... a
* ..... 4 .... 4a+b ....... a
* ..... 5 .... 5a+b ....... a
y' is the first difference = a
Difference in quadratic function sequences : y = a*x^2 + b*x + c
* x ......... y ...... y' ..... y"
* 0 ......... c
* 1 ..... a+b+c .... a+b
* 2 ... 4a+2b+c ... 3a+b ..... 2a
* 3 ... 9a+3b+c ... 5a+b ..... 2a
* 4 .. 16a+4b+c ... 7a+b ..... 2a
y" is the second difference = 2a (Note : second derivatve = 2a)
Difference in cubic function sequences : y = a*x^3 + b*x^2 + c*x + d
* x ............. y ......... y' ..... y" .... 3rd diff
* 1 ....... a+b+c+d ..... a+b+c
* 2 .... 8a+4b+2c+d ... 7a+3b+c ... 6a+2b
* 3 ... 27a+9b+3c+d .. 19a+5b+c .. 12a+2b .... 6a
* 4 .. 64a+16b+4c+d .. 37a+7b+c .. 18a+2b .... 6a
* 5 . 125a+25b+5c+d .. 61a+9b+c .. 24a+2b .... 6a
The 3rd difference = 6a (Note : 3rd derivatve = 6a)
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