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.

Growth True growth Increase in weight due to protein and bone deposition and fat deposition which is part of the developmental process Does not include excess fat deposition Don't know Why growth starts How it is regulated Why it stops DNA is ultimate regulator Cell size and number...