Explanation Obtain the formula for 2 2 + 4 2 + 6 2 + ⋯ + ( 2 n ) 2 as shown below. 2 2 + 4 2 + 6 2 + ⋯ + ( 2 n ) 2 = ( 2 × 1 ) 2 + ( 2 × 2 ) 2 + ( 2 × 3 ) 2 + ⋯ + ( 2 × n ) 2 = ( 2 2 × 1 2 ) + ( 2 2 × 2 2 ) + ( 2 2 × 3 2 ) + ⋯ + ( 2 2 × n 2 ) = 2 2 ( 1 2 + 2 2 + 3 2 + ⋯ + n 2 ) It is known that, 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 . Therefore, the formula for 2 2 + 4 2 + 6 2 + ⋯ + ( 2 n ) 2 becomes, 2 2 + 4 2 + 6 2 + ⋯ + ( 2 n ) 2 = 2 2 ( n ( n + 1 ) ( 2 n + 1 ) 6 ) = 2 3 n ( n + 1 ) ( 2 n + 1 ) By method of mathematical induction, verify whether the formula 2 3 n ( n + 1 ) ( 2 n + 1 ) is correct or not as follows. Substitute n = 1 in 2 3 n ( n + 1 ) ( 2 n + 1 ) and obtain s 1 as shown below. s 1 = 2 3 ( ( 1 ) ( 1 + 1 ) ( 2 ( 1 ) + 1 ) ) = 2 3 ( 2 ) ( 3 ) = 4 = 2 2 Thus, the formula is correct for n = 1 . Now assume the formula is correct for n = k . Then, s k = 2 3 k ( k + 1 ) ( 2 k + 1 )

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Chapter 9.2, Problem 9E

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Chapter 9 Solutions

Discrete Mathematics

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Chapter 9 Solutions