Chapter 9 equeces, eries, ad Probability ectio 9. Mathematical Iductio ectio Objectives: tudets will lear how to use mathematical iductio to prove statemets ivolvig a positive iteger, recogize patters ad write the th term of a sequece. tudets will fid sums of powers of itegers ad fid fiite differeces of sequeces. Math Aalysis.0: tudets ca give proofs of various formulas by usig the techique of mathematical iductio. Example Fid P + for each give P. a b c P + P + + + + + + ( 6 P ( + 6 6 6 P + + + + + ( + (( + + ( ( + P 6 ( + + + + P + 6 6 + + 6 ( + 6( + + d P ( + + + ( ( + P ( + + + + + ( + [What you practiced above is ecessary for what we will lear today, which is a way to prove formulas are true.] I. Itroductio The Priciple of Mathematical Iductio: o Let P be a statemet ivolvig the positive itegers. If. P is true, ad. Aytime P is true P + is also true for every positive iteger, the P must be true for all positive itegers.
Example Use mathematical iductio to prove the followig formula: ( + + + + +. ( +. is true. Therefore, the formula is valid for.. Assume that the formula is valid for is true, ad show that + is also true. ( + + + + + + ( + ( + + + + + + ( + + ( + Now show that the right side is "true" too. ( + ( + + + + + + + ( + ( + + is the ext term, which is what you get whe you add ( +. The ext term,, ca be gotte by addig ( + to both sides. Fid the commo deomiator so that you ca add the fractios together. Factor ad show that the ed result is the same as the give: ice + ( + ( + (( + + ( + ( + Example Use mathematical iductio to prove 7 9 ( ( + + + + + +.. how that the statemet is true for. ( +.. Assume that 7 9 ( ( + 7+ 9+ + ( + ( + + 7+ 9+ + ( + + ( + ( + ( + + ( + ( + ( + + ( + + They are the same, so the proof is doe! + + + + + + is true. The show that + is also true. + + + + 6 + + + ( ( Add the ext term i to both sides of the equatio. Remember lie what we practiced i Ex? ( ( (( ( ( If + The s + + + + + +
Example Use mathematical iductio to prove the followig formula: ( + + + + +. ( +.. Therefore the formula is valid for.. Assume that the formula is valid for, ad the show that it is valid for +. + + + + ( + ( ( + ( + ( ( + ( + ( + + + + + + + ( + + ( + + + + + ( ( + + + + ( + + + Therefore, the formula is valid for all positive itegers. The ext term,, ca be gotte by addig ( + to both sides. Fid the LCD so we ca add the two fractios together. Let s try somethig more elegat. ee how both fractios have( +? Factor out ( + from both. This ed result is the same as the give: ( + ( + ( + + + + ( ( (
Example Use mathematical iductio to prove the followig formula: ( ( ( (. how the statemet is true for. (. Assume that ( ( ( ( ( ( + + + + +. We do t ever really show that it s ( ( +. true for,,, but we could. For istace, whe : ( ( + + + + + ( + ( 0+ We have to show that it s also true for +. ( ( + ( ( ( ( + ( + ( + ( + + ( ( ( + ( + ( + ( + + ( + ( + (( + + ( + (( + ( ( + ( + ( + Let's try doig the more "elegat" way agai. What do the two fractios have i commo? ( + + Agai, compare the ed result with the give: + ( + ( ( + ( + ( + ( + (( + (( + + ( + ( ( + They are the same. Woo hoo!
DAY TWO II. Patter Recogitio ometimes we do ot have a formula that applies. o we the have to produce (come up with a formula through patter recogitio ad the prove it by mathematical iductio. Why? How do you ow the formula will always wor? You eed to prove it. The way to do so is by Iductio. Example Fid the formula for the fiite sum ad prove it by mathematical iductio., 7,,,,. Let s first list come partial sums to see if we ca determie the patter: + 7 0 + 7+ 7 + 7 + + 6 9 ( icreasig odd umbers ( + Now this is just a guess. ice we came up with this patter o our ow, we eed to prove that it will wor for ALL itegers.. how that it is true for. s ( ( + (. + 7+ + + + ( ( + + 7+ + + + ( + ( ( + ( + + ( ( + ( + + ( ( + + + ( +. Assume that + + + + + ( ( + + This may ot loo lie ( ( ( ( + + + + + + from above, but it is...try looig at it thisway Compare the results to the give: ( ( + ( + ( [ ] ( ( + + + + + + + We foud ad proved a ew formula!
Example Fid a formula for the th partial sum of a ( + ( + We eed to first fid some partial sums to ote the patter:, ad the prove by mathematical iductio. ( + 8 + + ( + + + + + + 6 6 0 0 0 0 6 8 + 7 8 + + + + + 6 67 67 6 7 67 + It appears that ( + ( (. o we eed to prove this formula wors by mathematical iductio.. We already ow is valid (from above.. We assume the formula is valid for, ad the show it is valid for +. + + + + 6 + + + + + ( + + ( + ( + ( + + ( + ( + ( ( ( + + + + + + 6 ( + ( + ( + + ( + + ( + ( + + ( + + + ( + ( + + ( + + + ( + ( + ( + + + + + + + ( ( ( + ( + + ( + ( This result is the same as the give: ( + + + ( + ( + + + ( + + Therefore, the formula is valid for all positive itegers. ( Add the ext term to both sides to get.
III. ums of Powers of Itegers The followig formulas are called the ums of Powers of Itegers ( + + + + + + + + + ( + ( + 6 ( + + + + + + + + + 0 + + + + ( + ( + ( + ( + ( + Example Fid the followig sums. a. 0 ( + ( + 0 0 0 0 0,8 b. 8 i ( + ( + ( + 89 ( ( 7( 6 + 8 i i i i i 877 0 0 c. ( i i i i i i i ( ( ( i i+ i+ i i+ 6 ( 6 ( 6 ( 6 6 7 90 0 0 0 d. ( 7 7 ( ( ( + + + 7 6 ( ( ( 0 0 7 6 0 70 880