Prove that the formula 13+33+53+...+(2n−1)3=n2(2n2−1) is true for all natural numbers n.
By using mathematical induction,
Let P(n) denote the statement 13+33+53+...+(2n−1)3=n2(2n2−1).
Then, we need to show that P(1) is true. So,
13=(1)2(2(1)2−1)1=(2−1)1=1
Thus, we prove the first principle of the mathematical induction. More over, assuming that P(k) is true, then
13+33+53+...+(2k−1)3=k2(2k2−1)
Now, by showing P(k+1), we have
13+33+53+...(2k−1)3+[2(k+1)−1]3=(k+1)2[2(k+1)2−1]13+33+53+...(2k−1)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+...+(2k−1)3]+[[2(k+1)−1]3]Group the first k terms=k2(2k2−1)+[2(k+1)−1]3Induction hypothesis=k2(2k2−1)+[2k+1]3Expand=k2(2k2−1)+8k3+12k2+6k+1Combine like terms=2k4−k2+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.
Saturday, October 7, 2017
College Algebra, Chapter 9, 9.5, Section 9.5, Problem 10
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