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Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2. n(5n 3) Foreveryintegernz 1,1+6+11+16+--- +(5n4): 2 Proof (by mathematical induction): Let P(n) be the equation n(5n 3) . 1+6+11+16+v--+(5n4)= 2 We will show that Phil) is true for every integer n 2 1. Show that P(1) is true: Select P(1) from the choices below. OP(1)=1'[5'21_3) O P(1)=1 01:1'(5'1_3) 2 O 1+(5-14)=1-(5-13) The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k 2 1, and suppose that P(k) is true. The lefthand side of P(k) is and the righthand side of P(i() is [The inductive hypothesis states that the two sides of P(k) are equai.] We must show that P(k + 1) is true. P(k + 1) is the equation 1 + 6 + 11 + 16 + + (5(k + 1) 4) = |:| .After substitution from the inductive hypothesis, the lefthand side of PU: + 1) becomes + (5(k + 1) 4). when the lefthand and righthand sides of PU: + 1) are simplified, theyr both can be shown to equal |:| . Hence P(k + 1) is true, which completes the inductive step. [Thus both the basis and the inductive steps have been proved, and so the promf by mathematicai induction is compiete.]