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Senior High Math : Home work by keywords
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Sequences : 1, 8, 27, 64, 125, ....
* Example 1 : 2nd sequence of quadratic function
* Example 2 : 3rd sequence of Cubes function
* Example 3 : Series
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Sequences : 1, 4, 7, 13, 23, ....
* Example 1 : Find T(n)
* Example 2 : Find S(n)
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Sequences : 1, 4, 10, 20, 35, ....
* Example 1 : Find T(n)
* Example 2 : Find S(n)
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Sequence of numbers in cube patterns
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Sequence of numbers in square patterns
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Sequence of numbers in triangular patterns
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Sequence : Fibonacci's sequence T(k+2) = T(k) + T(k+1)
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Sequence of triangular number
* Example 1 : Prove that Sum[n*(n+1)/2] = n*(n+1)*(n+2)/6
* Example 2 : Prove that Sum[C(n+1,2)] = C(n+2,3)
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Series of functions
- CA 18 01 | - Series of arctan(x)
- CA 18 02 | - Seires of ln(1+x)
- CA 18 03 | - Series of ln(1 - x)
- CA 18 04 | - series of e^x
- CA 18 05 | - Series of e^(-x)
- CA 18 06 | - Series of sinh(x)
- CA 18 07 | - series of cosh(x)
- CA 18 08 | - Series of sin(x)
- CA 18 09 | - Series of cos(x)
- CA 18 10 | - Series of arcsin(x)
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Series of functions
- e^x = 1 + x + (x^2)/(2!) + (x^3)/(3!) + ...
- sin(x) = +x - (x^3)/(3!) + (x^5)/(5!) - ...
- cos(x) = +1 - (x^2)/(2!) + (x^4)/(4!) - ...
- sinh(x) = x + (x^3)/(3!) + (x^5)/(5!) + ...
- cosh(x) = 1 + (x^2)/(2!) + (x^4)/(4!) + ...
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Series of pi
- 1. Series : arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ....
- 2. Series : (pi)^2 = 6*(1 + 1/(2^2) + 1/(3^2) + ......)
- 3. Series : (pi)^2 = 8*(1 + 1/(3^2) + 1/(5^2) + ......)
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Series of pi
- 1. (pi)^2 = 8*(Sum[1/(2*n-1)^2])
- 2. (pi)^2 = 8*(Sum[1/(2*n-1)^2] - Sum[(6*n-3)^2])
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Series of csc(z)^2
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Series : 1 + 4 + 9 + 16 + 25 + .... + n²
* Example 1 : Prove that Sum[n^2]=n*(n+1)*(2*n+1)/6
* Example 2 : Squares in squares
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Series : 1 + 8 + 27 + 64 + 125 + .... + n³
* Example 1 : Prove that Sum[n^3]=(n*(n+1)/2)^2
* Example 2 : Cubes in cubes
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Series : Special
- 1. S = 1/(1*2) + 1/(2*3) + ..... + 1/(n*(n+1))
- 2. S = 1/(3*5) + 1/(5*7) + ...... + 1/((2*n-1)*(2*n+1))
- 3. S = 1/(1*2*3) + 1/(2*3*4) + 1/(3*4*5) + ... + 1/(n*(n+1)*(n+2)
- 4. S = (1-1/4)*(1-1/9)*(1-1/16)*.....*(1-1/(n^2)) where n GT 2
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Series : Special
- 1. S(n) = Sum[1/(n*(n+1))]
- 2. S(n) = Sum[1/((2*n-1)*(2*n+1))]
- 3. S(n) = Sum[1/(n*(n+1)*(n+2))]
- 4. S(n) = Sum[(1-1/(n^2))] where n GT 2
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Series : Sum[n^2] = n*(n+1)*(2*n+1)/6
- 1. Prove by observation
- 2. prove by using sum[C(n+1,2)] = C(n+2,3)
- 3. Prove by induction
- 4. Prove by Sum[(x+1)^3 - x^3] = (x + 1)^3 - 1
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Series : Sum[n^3] = (n*(n+1)/2)^2
- 1. Prove by observation
- 2. prove by using sum[C(n+2,3)] = C(n+3,4)
- 3. Prove by induction
- 4. Prove by Sum[(x+1)^4 - x^4] = (x + 1)^4 - 1
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Series : Sum[n^4] = (n*(6*n^4 + 15*n^3 + 10*n^2 - 1)/30
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Series : Pascal triangle and sequences
- Prove that Sum[C(n+1,2)] = C(n+2,3)
- Prove that Sum[C(n+2,3)] = C(n+3,4)
- Prove that Sum[C(n+3,4)] = C(n+4,5)
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Sine function : Definition and outlines
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Sine function : Summary
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Sine function : Identities in triangle ABC
- 1. sin(A) + sin(B) + sin(C) = 4*cos(A/2)*cos(B/2)*cos(C/2)
- 2. sin(2*A) + sin(2*B) + sin(2*C) = 4*sin(A)*sin(B)*sin(C)
- 3. sin(A)^2 + sin(B)^2 + sin(C)^2 = 2 + 2*cos(A)*cos(B)*cos(C)
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Sine function : Sine law
- 1. Area of triangle = a*b*c/(4*R)
- 2. Area of triangle = 2*(R^2)*sin(A)*sin(B)*sin(C)
- 3. Solve triangle if SAA is goven
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Sine function : Five points method
- Sketch y = Sqr(3)*cos(x) + sin(x)
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Sin(18) : Proof and application
- 1. Find sin(9)
- 2. Find sin(36)
- 3. Find cos(9), cos(18), cos(36)
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Sin(30) : Proof and application
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Sin(A) = 2*Sqr(s*(s-a)*(s-b)*(s-c))/(b*c)
- Area of triangle = Sqr(s*(s-a)*(s-b)*(s-c))
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Square : Change to an equal area triangle
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Squares in square patterns
- 1. Sketch a squares in square for 4^2
- 2. Sketch a squares in square for 5^2
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Star Functions : R = a + b*sec(p*A/q)^M
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Symmetrical matrix order 5
- 1. Find element row 1 and column 5 for power 3 of the matrix
- 2. Find element row 1 and column 5 for power 4 of the matrix
- 3. Find element row 1 and column 5 for power 5 of the matrix
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Synthetic division
- Express x^3 - 6*x^2 + 11*x - 6 in factor form
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