2. Prove that if n is even, then 3n + 1 is odd

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

A:

Q: Show that for every integer n, n³ + 2 is not divisible by 4.

A: Here, we use rules of divisibility.

Q: 3n > n for every positive integer n.

A: Now, since we assumed 3k>k, we can substitute k with 3k in the inequality: 3k+1>3k Since ,…

Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

A:

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: (2) Use a proof by ca. If is int er

A:

Q: Prove, by induction, that n + 3n2 + 2n is divisible by 6, for n 2 0.

A: In this problem I have use mathematical induction method two times please read step by step.

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

A:

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: п(п + 1)(2n + 1) 2. Prove that 12 + 22 + 32 + 4?+ ...+n2 for every positive integer n. 6.

A: Here we have to prove that       12+22+32+...+n2 = n(n+1)(2n+1)/6    for every positive integer n .