Disc-a7

# Disc-a7 - 2 or in general 1 3 5&&& 2 n ± 1 = n 2...

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1. Show by mathematical induction that 1 + 3 + 3 2 + + 3 n = 3 n +1 ± 1 2 for every positive integer n . The base case is when n = 1 , in which case the equation is 1+3 = 3 2 ± 1 2 , which is true. For the induction step, we assume that the equation holds for n = k : 1 + 3 + 3 2 + + 3 k = 3 k +1 ± 1 2 Add 3 k +1 to both sides 1 + 3 + 3 2 + + 3 k + 3 k +1 = 3 k +1 ± 1 2 + 3 k +1 = 3 k +1 + 2 3 k +1 ± 1 2 = 3 3 k +1 ± 1 2 = 3 k +2 ± 1 2 which is the equation when n = k + 1 . 2. n odd numbers is n 2 . For example , 1+3+5+7 = 4
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Unformatted text preview: 2 , or, in general , 1+3+5+ & & & +2 n ± 1 = n 2 . The base case is when n = 1 , in which case the equation is 1 = 1 2 , which is true. For the induction step, we assume that the equation holds for n = k : 1 + 3 + 5 + & & & + 2 k ± 1 = k 2 Add 2 k + 1 to both sides 1 + 3 + 5 + & & & + (2 k ± 1) + (2 k + 1) = k 2 + 2 k + 1 = ( k + 1) 2 which is the equation when n = k + 1 ....
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