Problem 20

Find the last digit of 43⁴³.

Problem 19

Find the last digit of 2⁹⁹⁹.

Problem 12

Prove that for all natural numbers n, the formula

 is true.

Problem 11

Prove that for all natural numbers n, the formula

 is true.

  

Problem 10

 Prove that for all natural numbers n, the formula 1 + 3 + 5 +…+ (2n - 1) =  n² is true.

Problem 4

The recursive formula for a sequence {u_n} is written in the form:

u₁ = 2, 
u_n+1 = 3u_n, n ≥ 1

Write the general explicit formula.

Problem 3

Prove that the following equality holds for all natural numbers n.

Problem 2

Prove that aⁿ – bⁿ = (a – b) (a^{n – 1} + a^{n – 2}.b +… +a.b^{n – 2} + b^{n – 1}) for all natural numbers n > 1.