MATH135 Study Guide - Midterm Guide: Summation, Divisor

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Wentang Kuo

Document Summary

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)