MATH135 Study Guide - Midterm Guide: Summation, Divisor
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MATH135 Full Course Notes
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The sum notation, denoted as is the sum of all numbers within the range from the bottom of the sigma notation to the top of the sigma notation. 10(cid:80) i=1 i2 = 12 + 22 + 32 + + 102 i: summation principles. Iii: n(cid:88) i=m cxi = c xi (xi yi) i=m n(cid:88) xi n(cid:88) n(cid:88) i=m i=m n(cid:88) n(cid:88) n+k(cid:88) i=m yi = xi = xj k i=m j=m+k ii: principle of mathematic induction. Prove: 12 + 22 + + n2 = n(cid:80) i=1. = 1 k(cid:88) i=1 k+1(cid:88) i2 = k(k + 1)(2k + 1) 6 k(cid:88) i2 = i2 + (k + 1)2. Inductive hypothesis: let"s assume that the statement p (k) is true for some k 1, k n. Inductive conclusion: let"s show that p (k + 1) is true i=1 i=1 k(k + 1)(2k + 1) By principles of mathematics induction, p (n) is true for all n n (cid:4)