3.6 +6.9+9.12+....+3n(3n+3) = 3n(n+1) (n+2)

for n=1, P(1) = 3(1+1)(1+2)
= 3*2*3 = 18
therefore P(1) is true
for n = k.    P(k) = 3k(k+1)(k+2)        assume p(k) to be true.
P(k+1) = p(k) + (k+1)th term
 3k(k+1)(k+2) + 3(k+1)(3k+4)
 3(k+1) [ k(k+2) + 3k+4 ]
 3(k+1) [ k2+5k+4] 
 3 ( k+1 ) ( k+2 ) ( k+3) = RHS
Thus by the principle of mathematical induction, the statement P (n) is true for all .

 

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Prove by induction that 3^n>n for any natural number n.
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