(Solved): plsase solve 1. \( (1+1+1+1+4=8 \) points) Suppose you wish to use mathematical induction to prove t ...

1. \( (1+1+1+1+4=8 \) points) Suppose you wish to use mathematical induction to prove that: \( 1 \cdot 1 !+2 \cdot 2 !+3 \cdot 3 !+\cdots+n \cdot n !=(n+1) !-1 \) for all \( n \geq 1 \). (a) Write \( P(1) \) (b) Write \( P(5) \). (c) Write \( P(k) \). (d) Write \( P(k+1) \) (e) Use mathematical induction to prove that \( \dot{P}(n) \) is true for all \( n \geq 1 \).