Prove each statement i using mathematical induction. Do

Chapter 5, Problem 6E

(choose chapter or problem)

Prove each statement i using mathematical induction. Do not derive them from Theorem 1 or Theorem 2.Theorem 1Sum of the First n IntegersFor all integers n ? 1, Proof (by mathematical induction):Let the property P(n) be the equation Show that P(1) is true:To establish P(1), we must show that But the left-hand side of this equation is 1 and the right-hand side is also. Hence P(1) is true.Show that for all integers k? 1, if P(k) is true then P(k + 1) is also true:[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ? 1.That is:] Suppose that k is any integer with k ? 1 such that [We must show that P(k + 1) is true. That is:] We must show that or, equivalently, that [We will show that the left-hand side and the right-hand side of P(k + 1) are equal to the same quantity and thus are equal to each other.]The left-hand side of P(k + 1) is And the right-hand side of P(k + 1) is Thus the two sides of P(k + 1) are equal to the same quantity and so they are equal to each other. Therefore the equation P(k + 1) is true [as was to be shown].[Since we have proved both the basis step and the inductive step, we conclude that thetheorem is true.]Theorem 2Sum of a Geometric SequenceFor any real number r except 1, and any integer n ? 0, Proof (by mathematical induction):Suppose r is a particular but arbitrarily chosen real number that is not equal to 1, and let the property P(n) be the equation We must show that P(n) is true for all integers n ? 0. We do this by mathematical induction on n.Show that P(0) is true:To establish P(0), we must show that The left-hand side of this equation is r0 = 1 and the right-hand side is also because r1 = r and r = 1. Hence P(0) is true.Show that for all integers k? 0, if P(k) is true then P(k+1) is also true:[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ? 0. That is:]Let k be any integer with k ? 0, and suppose that [We must show that P(k + 1) is true. That is:] We must show that or, equivalently, that [We will show that the left-hand side of P(k + 1) equals the right-hand side.]The left-hand side of P(k + 1) is which is the right-hand side of P(k + 1) [as was to be shown.][Since we have proved the basis step and the inductive step, we conclude that the theorem is true.]ExerciseFor all integers n ? 1, 2 + 4 + 6 + …+ 2n = n2 + n.