solve 28 step by step 28 1×1!+2×2!+3×3!+ +n×n!=n+1!-1 for all N∈N - Maths - Principle of Mathematical Induction

Question

solve 28 step by step

28 1 × 1 ! + 2 × 2 ! + 3 × 3 ! + + n × n !
= n + 1 ! - 1   for   all   N ∈ N

Vijay Kumar Gupta · Accepted Answer

Dear student,

Let Pn:1×1!+2×2!+3×3!+
+n×n!=n+1!-1for n=1, we have:LHS: 1×1!=1RHS: 1+1!-1=2!-1=2-1=1So, LHS=RHS for n=1Let Pn is true for n=k        Therefore,         1×1!+2×2!+3×3!+
+k×k!=k+1!-1        
1We will show that Pn is also true for n=k+1for n=k+1     Pk+1:1×1!+2×2!+3×3!+
+k×k!+k+1×k+1!=k+2!-1 Consider the LHS         1×1!+2×2!+3×3!+
+k×k!+k+1×k+1! ⇒   k+1!-1+k+1×k+1!      from equation 1 ⇒   k+1!+k+1×k+1!-1 ⇒   k+1!1+k+1-1 ⇒   k+1!k+2-1 ⇒   k+2!-1         n+1·n!=n+1!⇒   RHSThus for n=k+1, LHS=RHSTherefore, by principle of mathematical, statement Pn is true for all n∈N

Regards

solve 28 step by step 28 . 1 × 1 ! + 2 × 2 ! + 3 × 3 ! + . . . + n × n ! = n + 1 ! - 1 for all N ∈ N

solve 28 step by step

$28 . 1 \times 1! + 2 \times 2! + 3 \times 3! + . . . + n \times n! = (n + 1)! - 1 for all N \in N$