e. Prove the following for all n ≥ 1 using weak induction : n (a + b)² = Σ(n)a(n-¹ f² (n—r) fr recall : r=0 n! (n.) = k!(n - k)! k −

k-1 S₁=Σa+id=(2a+(k−1)d) i=0 Prove this using induction.

Prove by mathematical induction that Σ M i=1 = n?(n + 1)2 4 for all n ΕΝ.

Suppose you wish to use mathematical induction to prove that: SEE IMAGE answer the following: (c) Write P(k).(d) Write P(k + 1).(e) Use mathematical induction to prove that P(n) is true for all n ≥ 1

Use induction to show that, for any integer n ≥ 1: ∑ni=1    i * i! = (n+1)! - 1