Mathematical Induction Problem 1

 

Prove by Mathematical induction.

a) 2+4+6+…+2n= n² +n

solution

        let  the given statement  be p(n), then

p(n) = 2+4+6+…+2n= n²+n

when, n=1

L.H.S = 2      and   R.H.S = n²+n

                                        =1²+1

                                        =2

.: L.H.S = R.H.S

Let p(k) be true, then

P(k)= 2+4+6+…+2k = k²+k

Now,

           2+4+6+…+2k+2(k+1) =  k²+k+2(k+1)

                                                =k(k+1)+2(k+1)

                                                =k²+k+2k+2

                                               =k²+2k+1+k+1

                                               =(k+1)²+(k+1)

.:p(k+1) = 2+4+6+…+2k+2(k+1)  = (k+1)² (k+1)

.:p(k+1) is true , whenever p(k) is true

Hence, by the principle of mathematical induction p(n) is true.

 

 

 

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