Mathemtics Dictionary : Programs
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* Sample 1 : Y=((x-1)^3)/(2*x)
* Sample 2 : R=1+1*sin(9*A/4)^3
* Sample 3 : Magic number 1 to 33
* Sample 4 : Book Cover of Pictorial Mathematics
* Sample 5 : Tutorials
* What is Mathematic Dictionary ?
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Week 1 : Sum[n^3]=?
Example : Prove that 1^3+2^3+3^3+...+n^3 = (n*(n+1)/2)^2
Method 1 : By observation
1^3+2^3 = 9 = (1+2)^2
1^3+2^3+3^3 = 36 = (1+2+3)^2
1^3+2^3+3^3+4^3 = 100 = (1+2+3+4)^2
Since 1+2+3+4+.....+n = n*(n+1)/2
Proof complete
Method 2 : Use Sum[C(n+2,3)]=C(n+4,4)
C(n+2,3) = (n+2)*(n+1)*n/3!
C(n+3,4) = (n+3)*(n+2)*(n+1)/4!
Sum[(n+2)*(n+1)*n/3!] = (n+3)*(n+2)*(n+1)/4!
Expnand :
Sum[n^3+3*n^2+2*n] = 6*(n+3)*(n+2)*(n+1)*n/24
Hence Sum[n^3] = 6*(n+3)*(n+2)*(n+1)*n/24-3*Sum[n^2]-2*Sum[n]
Since Sum[n^2] = n*(n+1)*(2*n+1)/6 and Sum[n] = n*(n+1)/2
Hence Sum[n^3] = n*(n+1)*[(n+3)*(n+2)/4 - (2*n-1)/2 -1]
Simplify right hand side and proof complete
Method 3 : Use mathematical induction
n = 1 the sum is true
n = 2 the sum is true
n = k+1 should be true. i.e. S(k+1)=((k+1)*k+2)/2)^2 exists
n = k+1 S(k+1) = S(k)+(k+1)^3
Hence S(k+1) = (k*(k+1)/2)^2+(k+1)^3 = ((k+1)^2)*(k^2+4*(k+1))/4
and we have S(k+1) = ((k+1)*(k+2)/2)^2. Proof complete
Method 4 : Use (a+b)^3 = a^3+3*(a^2)*b+3*a*(b^2)+b^3
Sum[(x+1)^3] = x^3+3*x^2+3*x+1
Sum[(x+1)^3] = Sum[x^3]+3*Sum[x^2]+3*Sum[x]+Sum[1]
Hence Sum[x^3] = x^3+3*x^2+3*x+1-3*Sum[x^2]-3*Sum[x]-Sum[1]
Since Sum[x^2] = (x*(x+1)*(2*x+1))/6
Since Sum[x]=x*(x+1)/2
Since Sum[1]=x
Hence we have the proof by substitution and simplication
Home work :
a. Write all formulae used in above methods to note book.
b. Try to remember them
c. More series are given in Chapter 14
d. Reference : Program 14 13
Exercises :
a. Prove Sum[n] = (n*(n+1))/2 : See program 14 01
b. Prove Sum[n^2] = (n*(n+1)*(2*n+1)/6 : See Program 14 12
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Week 2 : Distance of point (x1,y1) to line A*x+B*y+C=0
Example : Find distance from (3,4) to x+y+1=0
Method 1 : Use formula
a. Formula : D=(A*x1+B*y1+C)/Sqr(A^2+b^2)
b. Where A=1 B=1 and C=1
c. Hence D=(1*3+1*4+1)/Sqr(3^2+4^2)
d. Hence D=8/5
Method 2 : Use basic methods
a. Find line perpendicular to A*x+B*y+C=0
b. The slope of the line A*x+B*y+C is m1=-B/A
c. The slope of the perpendicular line is m2=-1/m1
d. Find equation of the perpendicular line through (x1,y1)
e. Find intesection (x2,y2) of these 2 lines
f. Hence D=sqr((x2-x1)^2+(y2-y1)^2)
g. This method is good because we do not need remember formula
Home Work :
a. Use method 2 to prove that D=(A*x1+B*y1+C)/Sqr(A^2+B^2)
b. Referece : Program 38 06
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Week 3 :
Example : ?
Method 1 :
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