Question

Prove by Mathematical Induction that 1! + (2 × 2!) + (3 × 3!) + … + (n × n!) = (n + 1)! – 1

Answer 1

P(n) is the statement 

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

To prove for n = 1 

LHS = 1! = 1 

RHS = (1 + 1)! – 1 = 2! – 1 = 2 – 1 = 1

LHS = RHS ⇒ P(1) is true 

Assume that the given statement is true for n = k 

(i.e.) 1! + (2 × 2!) + (3 × 3!) + … + (k × k!) = (k + 1)! – 1 is true 

To prove P(k + 1) is true 

p(k + 1) = p(k) + t_{(k + 1)} 

P(k + 1) = (k + 1)! – 1 + (k + 1) × (k + 1)! 

= (k + 1)! + (k + 1) (k + 1)! – 1 

= (k + 1)! [1 + k + 1] – 1 

= (k + 1)! (k + 2) – 1 

= (k + 2)! – 1

= (k + 1 + 1)! – 1 

∴ P(k + 1) is true ⇒ P(k) is true, 

So by the principle of mathematical induction P(n) is true.