If Pr stands for rPr then the sum of the series 1 + P1 + 2P2 + 3P3 +

1.	If Pr stands for rPr then the sum of the series 1 + P1 + 2P2 + 3P3 + … + nPn is …..(a) Pn + 1 (b) Pn + 1 – 1 (c) Pn - 1 + 1(d) (n + 1)P(n - 1)
Answer» (b) P_{n + 1} – 1 1 + 1! + 2! + 3! + … + n! Now 1 + 1 (1!) = 2 = (1 + 1)! 1 + 1 (1!) + 2(2!) = 1 + 1 + 4 = 6 = 3! 1 + 1(1!) + 2(2!)+ 3(3!) = 1 + 1 + 4 + 18 = 24 = 4! 1 + 1(1!) + 2(2!) + 3(3!) ….+ n(n!) = (n + 1)! – 1 = ^{n + 1}P_{n + 1} – 1 = P_{n + 1} – 1

If Pr stands for rPr then the sum of the series 1 + P1 + 2P2 + 3P3 + … + nPn is …..(a) Pn + 1 (b) Pn + 1 – 1 (c) Pn - 1 + 1(d) (n + 1)P(n - 1)

Answer»

(b) P_{n + 1} – 1

1 + 1! + 2! + 3! + … + n!

Now 1 + 1 (1!) = 2 = (1 + 1)!

1 + 1 (1!) + 2(2!) = 1 + 1 + 4 = 6 = 3!

1 + 1(1!) + 2(2!)+ 3(3!) = 1 + 1 + 4 + 18 = 24 = 4!

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

= ^{n + 1}P_{n + 1} – 1 = P_{n + 1} – 1