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Asked by Muskan Last Modified
Answer Certainly! Let's prove the given statement using the Principle of Mathematical Induction.
Base Case: For n=1n=1, 1×1!=(1+1)!−1=2!−1=2−1=11×1!=(1+1)!−1=2!−1=2−1=1 So, the base case holds true.
Inductive Step: Assume the given statement is true for some arbitrary natural number kk. That is, assume 1×1!+2×2!+3×3!+⋯+k×k!=(k+1)!−11×1!+2×2!+3×3!+⋯+k×k!=(k+1)!−1
We need to show that the statement holds true for k+1k+1. So, let's consider: 1×1!+2×2!+3×3!+⋯+k×k!+(k+1)×(k+1)!1×1!+2×2!+3×3!+⋯+k×k!+(k+1)×(k+1)!
Now, we can rewrite (k+1)×(k+1)!(k+1)×(k+1)! as (k+1)!(k+1)!. So,
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