COT3100Induction02 - More Induction Examples Prove the...

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More Induction Examples Prove the following formula is true for all positive integers n. = - - - + = - n i n i n n i 1 1 2 1 2 ) 1 )( 1 ( ) 1 ( Use induction on n. Base Case. n=1. LHS = (-1) 0 1 1 = 1, RHS = (1(1+1)(-1) 0 )/2 = 1 Assume for an arbitrary value of n=k that = - - - + = - k i k i k k i 1 1 2 1 2 ) 1 )( 1 ( ) 1 ( Under this assumption, prove for n=k+1 that + = - + - - + + + = - 1 1 1 ) 1 ( 2 1 2 ) 1 )( 1 ) 1 )(( 1 ( ) 1 ( k i k i k k i 2 2 1 1 1 1 2 1 ) 1 ( ) 1 ( ) ) 1 ( ( ) 1 ( + - + - = - + = = - - k i i k k i k i i i = k(k+1)(-1) k-1 /2 + (-1) k (k+1)(k+1), using IH. = (k+1)(-1) k [-k/2 + k + 1] = (k+1)(-1) k [k/2 + 1] = (k+1)(-1) k (k + 2)/2 = (k+1)(k+2)(-1) k /2
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Some Algebra Rules . .. Laws of exponent and Logarithm: If a > 0, a x a y = a x + y ( a x ) y = a xy a x / a y = a x y a x = 1 / ( a x ) If b > 0 and b 1, log b ( xy ) = log b x + log b y log b ( x / y ) = log b x log b y log b ( x p ) = p log b x . Rules of inequalities: a > b a + c > b + c if c > 0, then a > b a c > b c if a > b and b > c a > c ; if a > b and c > d a + c > b + d . Useful algebra rules: ab = 0 a = 0 or b = 0 if bd 0, then a / b = c / d ad = bc ; ( a + b ) 2 = a 2 + 2 ab + b 2 ; ( a + b )( a b ) = a 2 b 2 .
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= n i i f 1 ) ( = = m i i f 1 ) ( + + = n m i i f 1 ) ( , if 1 m < n. = b
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COT3100Induction02 - More Induction Examples Prove the...

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