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Math Help - mathematical induction problem no.2

  1. #1
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    mathematical induction problem no.2

    mathematical induction problem no.2
    sorry for my laziness of not typing it!
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  2. #2
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    Quote Originally Posted by afeasfaerw23231233 View Post
    mathematical induction problem no.2
    sorry for my laziness of not typing it!
    I don't know if this is how you are to do it, but my first thought is that your series:
    2^3 + 6^3 + 10^3 + ~...~ + 38^3

    = 2^3(1^3 + 3^3 + 5^3 + ~...~ + 19^3)

    Now consider
    1^3 + 2^3 + 3^3 + ~...~ + 19^3 = \frac{19^2(19^2 + 1)}{4}

    (1^3 + 3^3 + 5^3 + ~...~ + 19^3) + (2^3 + 4^3 + 6^3 + ~...~ + 18^3) = \frac{19^2(19^2 + 1)}{4}

    (1^3 + 3^3 + 5^3 + ~...~ + 19^3) + 2^3(1^3 + 2^3 + 3^3 + ~...~ + 9^3) = \frac{19^2(19^2 + 1)}{4}

    And the second sum is just a sum of cubes, so
    (1^3 + 3^3 + 5^3 + ~...~ + 19^3) + 2^3 \left ( \frac{9^2(9^2 + 1)}{4} \right ) = \frac{19^2(19^2 + 1)}{4}

    1^3 + 3^3 + 5^3 + ~...~ + 19^3 = \frac{19^2(19^2 + 1)}{4} - 2^3 \left ( \frac{9^2(9^2 + 1)}{4} \right )

    And finally:
    2^3 + 6^3 + 10^3 + ~...~ + 38^3 = 2^3(1^3 + 3^3 + 5^3 + ~...~ + 19^3) =  2^3 \left [ \frac{19^2(19^2 + 1)}{4} - 2^3 \left ( \frac{9^2(9^2 + 1)}{4} \right ) \right ]

    -Dan
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  3. #3
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    Hello, afeasfaerw23231233!

    Here's another approach . . .


    Given: . 1^3 + 2^3 + 3^3 + \cdots + n^3\;=\;\frac{n^2(n+1)^2}{4} . for all positive integers n

    find the value of: . A \;= \;2^3 + 6^3 + 10^3 + \cdots + 38^3

    We have: . S \;=\;(2\cdot1)^3 + (2\cdot3)^3 + (2\cdot5)^3 + \cdots + (2\cdot19)^3

    . . . . . . . . . = \;(2^3\cdot1^3) + (2^3\cdot3^3) + 2^3\cdot5^3) + \cdots + (2^3\cdot19^3)

    . . . . . . . . . = \;8(1^3 + 3^3 + 5^3 + \cdots + 19^3)


    Let H \:=\:1^3 + 3^3 + 5^3 + \cdots + 19^3

    Let J \:=\:1^3 + 2^3 + 3^3 + \cdots + 20^3\:=\:\frac{20^2\!\cdot\!21^2}{4} \:=\:44,100
    Let K \:=\:2^3 + 4^3 + 6^3 + \cdots + 20^3 \:=\;2^3(1^3 + 2^3 + \cdots + 10^3) \:=\:8\cdot\frac{10^2\!\cdot\!11^2}{4} \:=\:24,200

    Hence: . H \;=\;J - K \;=\;44,100 - 24,200 \:=\:19,900


    Therefore: . S \;=\;8(19,900) \;=\;159,200

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  4. #4
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    thanks. i shouldn't make it as [1x4-2]^3+[2x4-2]^3+...
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