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Question

Prove using mathematical induction that for all n1

1+4+7+..+(3n-2)=n(3n1)2


A

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Solution

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For any integer n1,Pn be the statement that

1+4+7+..+(3n-2)=n(3n1)2

Base case–––––––––: The statement P1 says that

1=1(31)2.

Which is true.

Inductive step. Fix k1 , and supoose that Pk holds, that is,

1+4+7+...+(3k2)=k(3k1)2.

It remains to show that pk+1 holds, that is,

1+4+7+...+(3(k+1)2)=(k+1)(3(k+1)1)2.

1+4+7+...+(3(k+1)2)=1+4+7+...+(3(k+1)2)

= 1+4+7+...+(3k+1)

= 1+4+7+...+(3k2)+(3k+1)

=k(3k1)2+(3k+1)

= k(3k1)+2(3k+1)2

=3k2k+6k+22

=3k2+5k+22

=(k+1)(3k+2)2

= (k+1)(3(k+1)1)2

Pk+1holds.

thus, by the principle of mathematical induction for all n1,Pn holds.


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