Prove by induction: Prove that for all natural numbers k and a that 3((5*2^(2k+1) -1)/3 + a*4^k+1) is in the form of (

Please show all work, so I can understand! Thank you!

Suppose that a particular real number has the property that (x + (1/x)) is an integer. Use(strong) induction to prove that (x^n + (1/x^n)) is an integer for all natural numbers n.

Show that the system consisting of all elements of the form a+b√5, where a and b are any rational numbers, satisfies all the axioms for a field if the usual rules for addition and multiplication are used.

1. Prove that for all numbers of k, (5*2^(2k+1) - 1)/3 is an odd number.   2. Prove that for all natural numbers k, a (4^k - 1)/3 + 2*a*4^k is of the form 1 + 6a   3. Prove that for all natural numbers k, a (5*2^(2k+1) - 1)/3 + a*4^k is of the form 5 + 6a.   I need help in all of those three questions please. Proof by induction.

Show: For all integers and , if n-m is even then  n^3-m^3  is even

If a₁, a2, a3, a4 are the coefficients of any four consecutive terms in the expansion of (1 + x)", then prove that a1 az 2a2 + a₁ + a₂ a3 + a4 a₂ + az

Prove by induction that for positive integers n:  4|(5^(n+1)+ 3^(2n-1)).

Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

How many divisors does (1000000! + 1) have that are less than 1000000? Use Wilson's theorem