Continue the two sequences of numbers below and find an equation to calculate the n-th value: n 1 2 3 4 5 6 7 Equation an 2 5 10 17 26 37 bn 1 2 8 48 384 3840

Determine the numerical value of the following expression without the use of a calculator: (√2+(3^2 )^(1/4)+∑_(m=1)^3▒(1/m!-√m) ).(2^log_2⁡8 +1/2^3 -∏_(k=1)^8▒(1+1/k) )

Prove that n^5 − n is divisible by 5 for all positive integers n.

The drawing below shows two squares with side lengths a. One of the squares is rotated by 45◦ and placed in the centre of the second square. Find the surface area of the rotated square not overlapping the non-rotated square (grey area).