Explanation Consider the values for triangular number. Formula of nth triangular number is, n 2 ( ( 1 n ) − ( − 1 ) ) Substituting the values of n = 1 , 2 , 3 , 4 , 5 , 6 , 7 1st triangular number is, 1 2 ( 1 − ( − 1 ) ) = 1 2 nd triangular number is, 2 2 ( 2 − ( − 1 ) ) = 3 3 rd triangular number is, 3 2 ( 3 − ( − 1 ) ) = 6 4 th triangular number is, 4 2 ( 4 − ( − 1 ) ) = 10 5 th trtiangular number is, 5 2 ( 5 − ( − 1 ) ) = 15 6 th triangular number is, 6 2 ( 6 − ( − 1 ) ) = 21 7 th triangular number is, 7 2 ( 7 − ( − 1 ) ) = 28 Now consider the values for square number. Formula for nth square number is, n 2 ( 2 n − ( 0 ) ) Substituting the values of n = 1 , 2 , 3 , 4 , 5 , 6 , 7 1 st square number is, 1 2 ( 2 − ( 0 ) ) = 1 2 nd square number is, 2 2 ( 2 × 2 − ( 0 ) ) = 4 3 rd square number is, 3 2 ( 2 × 3 − ( 0 ) ) = 9 4 th square number is, 4 2 ( 2 × 4 − ( 0 ) ) = 16 5 th square number is, 5 2 ( 2 × 5 − ( 0 ) ) = 25 6 th square number is, 6 2 ( 2 × 6 − ( 0 ) ) = 36 7 th square number is, 7 2 ( 2 × 7 − ( 0 ) ) = 49 Now consider the values for pentagonal number

1st Edition

ISBN: 9781259218859

Author: David Sobecki Professor, Brian A. Mercer

Publisher: McGraw-Hill Education

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Chapter 2.6, Problem 4A

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| Without evaluating the Legendre symbols, prove the following. (i) 1(173)+2(2|73)+3(3|73) +...+72(72|73) = 0. (Hint: As r runs through the numbers 1,2,. (ii) 1²(1|71)+2²(2|71) +3²(3|71) +...+70² (70|71) = 71{1(1|71) + 2(2|71) ++70(70|71)}. 72, so does 73 – r.)

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Chapter 2 Solutions