day26 & 27

Day26&amp 27 - -0 6(e.J<.press,Or> in nJ 11 ~ t~\d~ or dtAP:llY\~ voria:bte s.Vf ~Yl = I d 3-L S = 15 Y1 = I IUO:r.2n=/oo 99-1- J

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Unformatted text preview: ----------0. 6 (e.J<.press ,Or-> in nJ 11:: ~ t~(\d~ or dtAP:llY\~ voria:bte] s ..Vf ~Yl. = I +d-.+3+-L{+S = 15 Y1 = I IUO :r. . .2n=/oo +99-1-· J.,., .::.. 0..&&--fore..-;>.. 7l=-I 'n = . ~_/D_'_+__I o_~~I +~+ ,o() so ~ ~::: IOU(/Ot) /00 (>01) - -SDSO~ . L ~Y1 := ' .. How do n:::.J we.. ~now%'\..S \S ~lA.e ~ Math 111 Mathematical Induction Principh~ of Mathematical Induction Given a statement involving integers n4! 1. The stat4aments is true for all n 2= 1 if (1) the statement is true for n -1, and (2) the sllltement is shown to be true for n -k + 1 under the assumption that the statement is tJlle for n .. k. n n(n + 1)Example. Prove that ~ i = 2 1 1(1 + 1)~(1) Consider the statement above for n - 1. It is to be shown t.hat ~i -f:f 2 1 N ~ l( 1+ 1) Th {': th 'f ow .f::f i ::; 1:: 2 '. ere. ore, e statement IS true or n ~_"1._~ (2) It is now assumed that the statement is true for n -k; that i , t i = ~(k2+ 1) :/; / is assumed to be true. It must hown that the statement is true or n -+ 1; -;,,\.qk+l ~ ( ' 't t be h h . (k +1)[(k + 1) + 1.) (k + l)(k + 2)that. IS 1 mllS sown t at I::::: - ----• V ' 2 2 ~ "':( ~ \~~ k+l k 1,\ '7/# ---7 Todothis,COJ\sjder ~I = ~i +~ "f/'f-J( ~0 ~~ 'l .f::f .f::f.J-~t.-~<I' (~ Jr~ = k(k+l) +(k+l) [by assumption above] 2R ::: k(k+1)+2(k+l)g ~ 1-(R+0 IS -+-he:....
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This note was uploaded on 12/05/2011 for the course MATH 111 taught by Professor Bang during the Fall '08 term at Emory.

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Day26&amp 27 - -0 6(e.J<.press,Or> in nJ 11 ~ t~\d~ or dtAP:llY\~ voria:bte s.Vf ~Yl = I d 3-L S = 15 Y1 = I IUO:r.2n=/oo 99-1- J

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