Question

prove the following by the principle of mathematical induction. 28 number

Answer 1

1*1! + 2*2! .................n*n! = (n + 1)! -1. so this is to prove by mathematical induction
so the first step is to put n = 1

L.H.S = 1*1! = 1
R.H.S = (1+1)! - 1 = 2! -1 = 1
so the statement is true for n = 1

now let us assume that the given statement be true for n = k
so,
1*1! + 2*2! +.........k*k! = (k+1)! -1. ...............(1)

now we shall prove that n = k+1 is true whenever n=k is true

1*1! + 2*2! + ....................(k+1)*(k+1)! =
[(k+1)+1] - 1

1*1! + 2*2! .............(k+1)*(k+1)! = [k+2] - 1

consider L.H.S

1*1! + 2*2! + .........k*k! + (k+1)*(k+1)!
now from equation (1)
(k+1)! - 1 + (k+1)*(k+)!
(k+1)! + (k+1)*(k+1)! - 1. [rearrange the statement]

(k+1)!{1 + k+ 1} - 1
(k+1)!*(k+2) -1
(k+2)! - 1 [ (k+2)! = (k+2)*(k+1)! ]
=R.H.S
hence proved

:-)