Prove that `1+2+3+4........+N<1/8(2n+1)^2`

Explanation

`" Let " P(n) : 1+2+3 +....+ nlt1/8(2n+1)^(2)`
for n=1
`L.H.S. =1, R.H.S. =1/8 (2+1)^(2)=9/8`
`:. L.H.S. lt R.H.S. `
`rArr` P(n) is true for n=1
Let P(n) be true for n=K
`P(k) : 1 +2+3+….+klt 1/8 (2k+1)^(2) ……(1)`
`n=K+1`
`P(k+1) : 1+2+3 +.....+K+(K+1)`
` lt 1/8 (2k+1)^(2)+(K+1) " "["From eauation "(1)]`
`=((2k+1)^(2)+8(k+1))/(8)`
`=(4k^(2)+4k+1+8K+8)/(8) =(4k^(2)+12K+9)/(8)`
`=1/8 (2k+2)^(2) =1/8 {2(k+1)+1}^(2)`
`rArr` P(n) is also true for n=K+1
Hence from the principle of mathematical induction the given statement is true for all `n in N`