The sum of series 1+2+3+………n is less than

Let the given statement be P(n). P(n):1+2+3+…+n<18(2n+1)2Step I: For n =1,1<18(2⋅1+1)2⇒1<18×32⇒1<98 which is true. Step ll :Let it is true for n = k,1+2+3+…+k<18(2k+1)2---iStep lll: For n =k+1,(1+2+3+…+k)+(k+1)<18(2k+1)2+(k+1)   [using Eq. (i)] =(2k+1)28+k+11=(2k+1)2+8k+88=4k2+1+4k+8k+88=4k2+12k+98=(2k+3)28=(2k+2+1)28=[2(k+1)+1]28⇒1+2+3+…+k+(k+1)<[2(k+1)+1]28Therefore, P(k + 1) is true when P(k) is true. Hence, from the principle of mathematical induction, the statement is true for all natural numbers n.