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What is the value of 2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 2 × 1) + .................. + 9(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) + 2 ?

What is the value of 2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 2 × 1) + .................. + 9(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) + 2 ? Correct Answer 10!

Concept:

1.1! + 2.2! + 3.3! + 4.4! + ..................... + n.n! = (n+1)! - 1

Calculation:

2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 2 × 1) + ................. + 9(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) + 2

⇒ 2.2! + 3.3! + 4.4! + 5.5! + 6.6! + 7.7! + 8.8! + 9.9! + 2

⇒ (3 - 1).2! + (4 - 1).3! + (5 - 1).4! + (6 - 1).5! + (7 - 1).6! + (8 - 1).7! + (9 - 1).8! + (10 - 1).9! + 2

⇒ 3.2! - 2! + 4.3! - 3! + 5.4! - 4! + 6.5! - 5! + 7.6! - 6! + 8.7! - 7! + 9.8! - 8! + 10.9! - 9! + 2

⇒ 3! - 2! + 4! - 3! + 5! - 4! + 6! - 5! + 7! - 6! + 8! - 7! + 9! - 8! + 10! - 9! + 2

⇒ (3! + 4! + 5! + 6! + 7! + 8! + 9! + 10!) - (2! + 3! + 4! + 5! + 6! + 7! + 8! + 9!) + 2

⇒ 10! - 2! + 2

⇒ 10! - 2 + 2

⇒ 10!

∴ 2.2! + 3.3! + 4.4! + 5.5! + 6.6! + 7.7! + 8.8! + 9.9! + 2 = 10!

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