Q. 124.5( 12 Votes )

Prove that:
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Answer :

To prove: P(1, 1) + 2. P(2, 2) + 3 . P(3, 3) + … + n . P(n, n) = P(n + 1, n + 1) – 1


We know,





Take L.H.S.:


1. P(1, 1) + 2. P(2, 2) + 3. P(3, 3) + … + n . P(n, n)


= 1.1! + 2.2! + 3.3! +………+ n.n!


{ P(n, n) = n!}






= (2! – 1!) + (3! – 2!) + (4! – 3!) + ……… + (n! – (n – 1)!) + ((n+1)! – n!)


= 2! – 1! + 3! – 2! + 4! – 3! + ……… + n! – (n – 1)! + (n+1)! – n!


= (n + 1)! – 1!


= (n + 1)! – 1


{ P(n, n) = n!}


= P(n+1, n+1) – 1


= R.H.S


Hence Proved


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