Question1; Marks:7 Use Mathematical Induction to prove that , for all integers n

Question

Question1;                                                                                                           Marks:7

Use Mathematical Induction to prove that

, for all integers n 1.

Question2;                                                                                                           Marks:7

Prove by mathematical induction that   n2-1 is divisible by 8, when n is an odd positive integer.

Question3 ;                                                                                                         Marks:6

Prove that if m is even and n is odd, then m+n-2 is odd.

Question3 ;                                                                                                         Marks:6

Explanation / Answer

1 )   for   all integers n>_ 1      the probelm isnot given 2) since n is odd +ve integer==> n = 2m -1 where m is +veinteger given data is (n2 -1 ) = (2m-1)2 -1                                   = 4m2 - 4m + 1 - 1        we have to showthat ( 4m2 - 4m ) is divisible by 8 for all +ve  integers of m by mathematical induction if m =1                                       4(1) - 4(1) = 0/8 =0 ==> the given data istrue for m = 1 we assume that given data istrue for m = k   [ 4k2 - 4k]/8 =p ==> 4k2 - 4k = 8p ==> 4k2= 8p + 4k ---------(1) if m = k+1 4[k+1]2 - 4[k+1] =4k2 +8k +4 - 4k - 4                                  =8p +4k  +8k - 4k                               = 8p +8 k                                 = 8 [ p+k ]                                = 8[ +ve iteger ] which is divisible by 8     yhe given data is true for m = k+1 hence by priniciple of mathematical induction given data is true for all +ve integers of m 3) m is even ==> m = 2k  and n is odd ==> n = 2k-1 where k is+ve integer   given data is [ m+ n - 2] = 2k + 2k - 1-2                                           =  4k - 3    let   S = 4k -3    we have to show that S is odd if k = 1 ==> S = 4 -3 = 1 it isodd     S is true for 1 we assume is true for S is true for k =p S =[ 4p - 3 ] is odd                                                                             let k = p +1 S = 4( p +1) - 3       = 4p + 4 - 3               = ( 4p -3) +4 ( 4p - 3) is odd + 4(even) = odd integer the given data is true for k =p+1 also hence by priniciple of mathmatical induction S is true for all +ve integers ofk                                                    given data is [ m+ n - 2] = 2k + 2k - 1-2                                           =  4k - 3    let   S = 4k -3    we have to show that S is odd if k = 1 ==> S = 4 -3 = 1 it isodd     S is true for 1 we assume is true for S is true for k =p S =[ 4p - 3 ] is odd                                                                             let k = p +1 S = 4( p +1) - 3       = 4p + 4 - 3               = ( 4p -3) +4 ( 4p - 3) is odd + 4(even) = odd integer the given data is true for k =p+1 also hence by priniciple of mathmatical induction S is true for all +ve integers ofk                                                   

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