11-proof-by-induction-s-k-nn12-maths-for-all-positive-year-12-hsc-extension-1

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Proof by induction Σ k =n(n+1)/2 maths for all positive Year 12 hsc Extension 1

5^n-2^n is divisible by 3,

n^3+2n is divisible by 3,

8^n-1 is divisible by 7,

1/(1X2)+1/(2X3)+...+1/(n(n+1))=n/(n+1),

n^2 less than2n,

8+2x8+3x8+..+8n=,

1+3+5+7+...+(2n-1)=n^2 for all n in N,

sum k^2= 1/6n(n+1)(2n+1),

9^n-2^n id divisible by 7,

1+5+9+13+..+(4n-3)=½n(4n-2),

n Σ k =n(n+1)/2 ,

1^3+2^3+3^3+..+n^3= (n(n+1)/2)^2 n^2(n+1)^2/4,

10^1+10^2+10^3+ +10^n = 10/9(10^n-1),

derivative of x^-1 is ( (-1)^n n !)/ x^(n+1),

2^n greater or equal 1+n,

3^n less thatn (n+1)!,

8^n-1 divisible by 7

Tagged under: 5^-2^ divisible 3,^3+2n divisible 3,8^-1 divisible 7,1/(1X2)+1/(2X3)+...+1/((+1))=/(+1),^2 than2n,8+2x8+3x8+..+8n=,1+3+5+7+...+(2n-1)=^2 N,sum ^2= 1/6n(+1)(2n+1),9^-2^ id divisible 7,1+5+9+13+..+(4n-3)=½(4n-2), Σ =(+1)/2,1^3+2^3+3^3+..+^3= ((+1)/2)^2 ^2(+1)^2/4,10^1+10^2+10^3+ +10^ = 10/9(10^-1),derivative ^-1 ( (-1)^ !)/ ^(+1),2^ greater equal 1+,3^ thatn (+1)!

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