Question

solve itt with proper steps..
_principle of Mathematical Induction_

Answer 1

Let P ( n ) be the statement given by

P( n ) = 1 + 3 + 5 + 7.. + ( 2 n - 1 ) = n^2

For n= 1, P ( 1 ) = 1

LHS = 1

RHS = 1 ^2 = 1

Assume that P ( k ) is true for some positive integer k.

P ( k ) = 1 + 3 + 5 + 7.. + ( 2 k - 1 ) = k^2

Now we shall prove that P ( k + 1 ) is also true.

P ( k + 1 ) = 1 + 3 + 5 + 7.. + [ 2 ( k+ 1) - 1] = ( k + 1)
^2

P ( k + 1 ) = k ^2 + [ 2 ( k+ 1) - 1] = ( k + 1)
^2

P( k + 1 ) = k^2 + 2 k +2 - 1 = ( k + 1)
^2

P ( k + 1 ) = k ^2 + 2k +1 = ( k + 1)
^2


P ( k + 1 ) = k^2 + k + k + 1 = ( k + 1)
^2


P ( k + 1 ) = k ( k + 1 ) + 1 ( k + 1 ) = ( k + 1)
^2


P ( k + 1 ) = ( k+ 1) ( k + 1 ) = ( k + 1)
^2

P ( k + 1 ) = ( k + 1) ^2 = ( k + 1)
^2

LHS = RHS

P ( K + 1 ) is also true.

Hence, By PMI , the given statement is true for all positive integers.

Answer 2

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