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day26 &amp; 27 - -6 0(e.J<.press,Or-> i n n J 1 1 ~ t...

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------ ---- 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 ~
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Math 111 M at hematical Induction Princip h~ of Mat hema ti cal Induction Given a statement involving integers n 4! 1. The sta t4 ame nts 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) Consi der 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; -;,,\.q k+ l ~ ( ' 't t be h h . (k + 1)[(k + 1) + 1.) (k + l) (k + 2) that . IS 1 mllS sow n 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 assum pti on above] 2 R ::: k(k+1)+2(k+l) g ~ 1- (R+0 IS -+-he:.
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