The right-hand side of P(k) is [The inductive hypothesis states that the two sides of P(k) are equal.] (k+1)+1 (k+1)+1 We must show that P(k + 1) is true. The left-hand side of P(k + 1) is i· 2'. When the final term of i. 2' is written separately, the result is i- 2 = i• 2i + - The i= 1 right-hand side of P(k + 1) is After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes + (k + 2)2k + When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal + 2. 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 proof by mathematical induction is complete.] Need Help? Read It

The right-hand side of P(k) is [The inductive hypothesis states that the two sides of P(k) are equal.] (k+1)+1 (k+1)+1 We must show that P(k + 1) is true. The left-hand side of P(k + 1) is i· 2'. When the final term of i. 2' is written separately, the result is i- 2 = i• 2i + - The i= 1 right-hand side of P(k + 1) is After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes + (k + 2)2k + When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal + 2. 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 proof by mathematical induction is complete.] Need Help? Read It

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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EVALUATE THE FOLLOWING
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Show complete solution
Let q(n) be the sum of the squares of the first n odd numbers. For example: q(0) = 0 q(1) = 12 q(2) = 12 + 32 q(3) = 12 + 32 + 52 q(4) = 12 + 32 + 52 + 72 ⋮ What would be the closed-form expression for q(n), where n is any integer greater than or equal to 0? Your expression must work for any such n, not just those shown in the examples.
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Answer is 17.
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