Saturday, October 7, 2017

College Algebra, Chapter 9, 9.5, Section 9.5, Problem 10

Prove that the formula 13+33+53+...+(2n1)3=n2(2n21) is true for all natural numbers n.

By using mathematical induction,

Let P(n) denote the statement 13+33+53+...+(2n1)3=n2(2n21).

Then, we need to show that P(1) is true. So,


13=(1)2(2(1)21)1=(21)1=1


Thus, we prove the first principle of the mathematical induction. More over, assuming that P(k) is true, then

13+33+53+...+(2k1)3=k2(2k21)

Now, by showing P(k+1), we have


13+33+53+...(2k1)3+[2(k+1)1]3=(k+1)2[2(k+1)21]13+33+53+...(2k1)3+[2(k+1)1]3=(k+1)2[2k2+4k+1]


We start with the left side and use the induction hypothesis to obtain the right side of the equation:


=[13+33+53+...+(2k1)3]+[[2(k+1)1]3]Group the first k terms=k2(2k21)+[2(k+1)1]3Induction hypothesis=k2(2k21)+[2k+1]3Expand=k2(2k21)+8k3+12k2+6k+1Combine like terms=2k4k2+8k3+12k2+6k+1=2k4+8k3+11k2+6k+1Factor by using synthetic division=(k+1)2(2k2+4k+1)


Therefore, P(k+1) follows from P(k), and this completes the induction step.

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