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Math Help - strong induction property of sequence

  1. #1
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    strong induction property of sequence

    hello
    on the jpg attachment...how does
    2^(k-1)+2^(k-2)+2^(k-3) becomes 2^0+2^1++2^(k-3)+2^(k-2)+2^(k-1) ?

    thank you
    Attached Thumbnails Attached Thumbnails strong induction property of sequence-shot1.jpg  
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by cortef View Post
    hello
    on the jpg attachment...how does
    2^(k-1)+2^(k-2)+2^(k-3) becomes 2^0+2^1+…+2^(k-3)+2^(k-2)+2^(k-1) ?

    thank you
    they just added 2^0 + 2^1 + \cdots for the sake of using the sum of the geometric series. it is just manipulation, a trick we can use to our advantage. don't let this jar you too much, the proof is valid. in fact, a similar trick was used earlier in the proof, obviously you didn't have a problem with it then. when proving P(2), they claimed: a_2 = 3 \le {\color{red}1} + 3 = 4 = 2^2. where did that 1 come from? they just added it. does that make sense? of course, surely if we are less than 3 we would be less than 3 + 1 = 4. it is the same trick used here. 2^0 + 2^1 + \cdots will be a positive number. so surely, if we are less than 2^{k - 1} + 2^{k - 2} + 2^{k - 3} then we will be less than \underbrace{2^0 + 2^1 +  \cdots}_{\mbox{a positive number}} + 2^{k - 1} + 2^{k - 2} + 2^{k - 3}
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