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Question

Prove by the principle of mathematical induction that for all nN:
1+4+7+....+(3n2)=12n(3n1)

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Solution

The statement S1 : 1 = 12(31)2=1 is true.
Assume that Sk : 1 + 4 + 7 + . . . + (3k − 2) = k2(3k1)is true --(1)

and to prove that Sk+1 : 1 + 4 + 7 + . . . + (3(k + 1) − 2) = k+12(3(k+1)1) is true.
Observe that
LHS:
1 + 4 + 7 + . . . + (3(k + 1) − 2) = 1 + 4 + 7 + . . . + (3k +1)

= k2(3k1)+(3k+1) from (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 =RHS
Thus by the Principle of Math Induction Sn is true for all natural numbers n.

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