1+2+3+...............+n
<18(2n+1)2
For n=1
18(2n+1)2=98
98>1
Let P(n)=1+2+3+..............+n<18(2n+1)2
P(1) is true.
Let P(k) be true
⇒1+2+3+.........+k<18(2k+1)2
P(k+1)=1+2+3+.........+k+k+1
P(k+1)<18(2k+1)2+k+1
=18((2k+1)2+8(k+1))
=18(4k2+1+4k+8k+8)
=18(4k2+12k+9)
P(k+1)<18(2k+3)2=18(2(k+1)+1)2
P(k+1) is true
∴P(k+1) is true ∀n∈N