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Question
The value of 1+1.1!+2.2!+3.3!+......+n.n!
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Solution
S=1.1!+2.2!+3.3!+............nn!
rearrange
S=(2-1)1! +(3-1) 2! +(4-1) 3!+.......(n+1-1) n!
=2 1! -1 1! +3 2! -1 2! + 4 3 ! -1 3!...............(n+1) n! - 1 n!
=2! -1! +3! -2! +4! -3! +..............(n+1)! -n!
=(n+1)! -1
S=1.1!+2.2!+3.3!+............nn!
rearrange
S=(2-1)1! +(3-1) 2! +(4-1) 3!+.......(n+1-1) n!
=2 1! -1 1! +3 2! -1 2! + 4 3 ! -1 3!...............(n+1) n! - 1 n!
=2! -1! +3! -2! +4! -3! +..............(n+1)! -n!
=(n+1)! -1
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