CMI_2011_TestB[S]

CMI_2011_TestB[S] - CMI Class Test-B Sol Q1 Total 8 marks(a...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CMI: Class Test-B [ Sol ] [ Q1 ] [ Total: 8 marks ] (a) Prove, by mathematical induction, that 2 + 4 + 6 + ··· + 2 n = n ( n + 1 ) for all positive integers n . [ 5 ] (b) By using the identity ( k + 2 ) 3- k 3 = 6 k 2 + 12 k + 8, or otherwise, find 2 2 + 4 2 + 6 2 + ··· + 100 2 . [ 3 ] [ Sol ] (a) For n = 1, L.H.S. = 1 = R.H.S. ∴ The statement is true for n = 1. Assume that the statement is true for some integer k ≥ 1, i.e., 2 + 4 + 6 + ··· + 2 k ) = k ( k + 1 ) For n = k + 1, 2 + 4 + 6 + ··· + 2 ( k + 1 ) = k ( k + 1 ) z }| { 2 + 4 + 6 + ··· + 2 k +( 2 k + 2 ) = k ( k + 1 ) + 2 ( k + 1 ) = ( k + 1 )[( k + 1 ) + 1 ] ∴ The statement is true for n = k + 1. Hence, by the principle of M.I., the statement is true for all positive integers n . (b) For k = 2,4,6,...,100, we have              4 3- 2 3 = 6 ( 2 ) 2 + 12 ( 2 ) + 8 6 3- 4 3 = 6 ( 4 ) 2 + 12 ( 4 ) + 8 8 3- 6 3 = 6 ( 6 ) 2 + 12 ( 6 ) + 8 . . . 102 3- 100 3 = 6 ( 100 ) 2 + 12 ( 100 ) + 8 . Adding up these 50 equations, we have 102 3- 2 3 = 6 ( 2 2 + 4 2 + 6 2 + ··· + 100 2 ) + 12 ( 2 + 4 + 6 + ··· + 100 ) + 8 × 50 ∴ 2 2 + 4 2 + 6 2 + ··· + 100 2 = 1061200- 12 × 50 ( 51 )- 400 6 = 171700. [ ALT ] 2 2 + 4 2 + 6 2 + ··· + 100 2 = 4 ( 1 2 + 2 2 + 3 2 + ··· + 50 2 ) = 4 50 ( 50 + 1 )[ 2 ( 50 ) + 1 ] 6 = 171700 AD: College Mathematics I [ 2011-12 ] 1 CMI: Class Test-B [ Sol ] [ Q2 ] [...
View Full Document

This note was uploaded on 02/22/2012 for the course CHEM yscn0027 taught by Professor Drtong during the Fall '10 term at HKU.

Page1 / 6

CMI_2011_TestB[S] - CMI Class Test-B Sol Q1 Total 8 marks(a...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online