Question

`(1+(1)/1)(1+(1)/(2))(1+(1)/(3))......(1+(1)/n)`
`n(n+1)`

Answer 1

Let
`P(n): (1+1) (1+(1)/(2))(1+(1)/(3))......(1+(1)/(n)) =(n+1)`
for n=1
`L.H.S. =1+1=2`
`R.H.S. =1+1=2`
`:. L.H.S. =R.H.S.`
`rArr` P (n) is true for n=1
Let P (n) be true for n =K
`:. P(k) : (1+1)(1+(1)/(2))(1+(1)/(3))......(1+(1)/(k)) =K+1`
for n = K+1
`P(k+1) : (1+1)(1+(1)/(2))(1+(1)/(3))`
` ....(1+(1)/(k))(1+(1)/(K+1))`
`=(k+1)(1+(1)/(K+1))`
`=(k+1)((K+2)/(K+1))=K+2`
`rArr` P(n) is also true for n= k+1
Hence from the principle of mathematical induction P (n) is true for all natural numbers n .