Monday, February 20, 2017

College Algebra, Chapter 9, 9.5, Section 9.5, Problem 12

Prove that the formula 112+123+134+...+1n(n+1)=n(n+1) is true for all natural numbers n.

By using mathematical induction,

Let P(n) denote the statement 112+123+134+...+1n(n+1)=n(n+1).

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


112=1(1+1)12=12


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

112+123+134+...+1k(k+1)=k(k+1)

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


112+123+134+...+1k(k+1)+1(k+1)[(k+1)+1]=k+1[(k+1)+1]112+123+134+...+1k(k+1)+1(k+1)(k+2)=k+1k+2


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


=[112+123+134+...+1k(k+1)]+[1(k+1)(k+2)]Group the first k terms=kk+1+1(k+1)(k+2)Induction hypothesis=k(k+2)+1(k+1)(k+2)Get the LCD=k2+2k+1(k+1)(k+2)Expand the numerator=(k+1)2(k+1)(k+2)Factor=k+1k+2Simplify


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

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