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# lec1102 - Some Algebra Rules Laws of exponent and Logarithm...

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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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A couple of summation rules = n i i f 1 ) ( = = m i i f 1 ) ( + + = n m i i f 1 ) ( , if 1 m < n. = b a i i f ) ( = = b i i f 1 ) ( - - = 1 1 ) ( a i i f , if b a > 1. = n i i f 1 ) ( = = - + n i i n f 1 ) 1 (
Couple More Examples: 1) Prove = n i i H 1 = (n+1)H n – n, using induction. Note that H n = = n i i 1 1 Use induction on n>0. Base case: n=1. LHS = 1/1 = 1 RHS = (1+1)(1/1) – 1 = 1 Assume for some n=k, = k i i H 1 = (k+1)H k – k Under this assumption, we must prove the formula for n = k+1: + = 1 1 k i i H = (k+2)H k – (k+1) + = 1 1 k i i H = ( = k i i H 1 ) + H k+1 = (k+1)H k – k + H k+1 , using inductive hypothesis.

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