1. Prove that 1(1!) + 2(2!) + … + n(n!) = (n + 1)! – 1, for all n ∈ . 2. Prove that    +  +……..+  = 1- , for all n . 3. Prove that 1+ 2⋅ 2 + 3⋅ 2 2 +L+ n2 n−1 = (n −1)2 n +1, for all n ∈ . 4. Prove...

1. Prove that 1(1!) + 2(2!) + … + n(n!) = (n + 1)! – 1, for all n ∈
.

2. Prove that

 +
 +……..+
 = 1-
, for all n
.

3. Prove that 1+ 2⋅ 2 + 3⋅ 2²
+L+ n2ⁿ⁻¹
= (n −1)2ⁿ
+1, for all n ∈
.

4. Prove that 5²ⁿ
– 1 is a multiple of 8 for all n ∈
.