Question

1. 72n+23n-3.3n-1 is divisible by 25 for all nN

Answer 1

7²ⁿ + 2³ⁿ.3ⁿ⁻¹
= (7²)ⁿ + (2³)ⁿ⁻¹. 3ⁿ⁻¹
= 49ⁿ  + 8ⁿ⁻¹.3ⁿ⁻¹
= (49)ⁿ + (8.3)ⁿ⁻¹
= 49ⁿ + 24ⁿ⁻¹
= (50-1)ⁿ + (25-1)ⁿ⁻¹
= 50ⁿ -n.50ⁿ⁻¹ + ....+(-1)ⁿ⁻¹. n50 +(-1)ⁿ] + [25ⁿ⁻¹ - n.25ⁿ⁻²+.... + (-1)ⁿ⁻². (n-1).25 +(-1)ⁿ⁻¹]

divide the whole equation by 25
we get
(-1)ⁿ + (-1)ⁿ⁻¹ = 0