For all natural numbers n, statement
` p(n)= 1+2+3+ 4....+n = (n(n+1))/2` is true find p(n+1)

Options

A. `(n(n+1))/2`
B. `((n+1)(n+2))/2`
C. `(n(n-1))/2`
D. `((n+1)(n+1))/2`

Answer

B

Explanation

Let `P (n) : 1+2+3+…. +N`
`=1/2 n(n+1)`
For n=1
`L.H.S =1`
`R.H.S =1/2 .1 (1+1) =1/2 .1.2=1`
`L.H.S. =R.H.S`
Therefore p (n) is true for n=1
Let P (n) is true for n=K
`:. P (K) :1+2+3+….+K =1/2 K(k+1)`
Adding (k+1) on both sides
`1+2+3+.....+K+(K+1)`
`=1/2 K(k+1) +(K+1)`
`=1/2 (K+1) (K+2)`
`=1/2 (K+1){(K+1)+1}`
`rArr " "P (n)" is also true for" n=K +1 `
Hence by the principle of mathematical induction, given statement is true for all natural number 'n'