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Question
The value of 1+1.1!+2.2!+3.3!+.....+n.n! is
The value of 1+1.1!+2.2!+3.3!+.....+n.n! is
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Solution
The correct option is D (n+1)!
1+1.1!+2.2!+3.3!+.....+n.n!=1+n∑r=1r.r!=1+n∑r=1(r+1−1)r!=1+n∑r=1(r+1)r!−r!=1+n∑r=1(r+1)!−r!=1+(2!−1!)+(3!−2!)+(4!−3!)+.....+((n+1)!−n!)=(n+1)!
1+1.1!+2.2!+3.3!+.....+n.n!=1+n∑r=1r.r!=1+n∑r=1(r+1−1)r!=1+n∑r=1(r+1)r!−r!=1+n∑r=1(r+1)!−r!=1+(2!−1!)+(3!−2!)+(4!−3!)+.....+((n+1)!−n!)=(n+1)!
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