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    Write down the first five terms of the sequence and give the recursive definition.
    I'm not sure how to find the recursive definition.

    ur=r^2
    I worked the first 5 terms 1,4,9,16,25

    ur=1/2r(r+1)
    First 5 terms 1,3,6,10,15

    ur=1/6r(r+1)(2r+1)
    First 5 terms 1,5,14,30,55
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    (Original post by osayukiigbinoba)
    Write down the first five terms of the sequence and give the recursive definition.
    I'm not sure how to find the recursive definition.

    ur=r^2
    I worked the first 5 terms 1,4,9,16,25
    Can you see that if you take the previous number and add a odd number to it you get the next term in the sequence?

    So, for example
    1 + 3 = 4
    4 + 5 = 9
    9 + 7 = 16
    16 + 9 = 25
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    (Original post by Zacken)
    Can you see that if you take the previous number and add a odd number to it you get the next term in the sequence?

    So, for example
    1 + 3 = 4
    4 + 5 = 9
    9 + 7 = 16
    16 + 9 = 25
    Yeah that makes sense thank you. My book says the answer is ur+1=ur+2r+1
    So does the 2r+1 represent the odd sequence you mentioned?
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    (Original post by osayukiigbinoba)
    Yeah that makes sense thank you. My book says the answer is ur+1=ur+2r+1
    So does the 2r+1 represent the odd sequence you mentioned?
    Yep. Try it out for r=1, 2, 3, 4, \cdots and see if it generates the sequence you want.
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    (Original post by Zacken)
    Yep. Try it out for r=1, 2, 3, 4, \cdots and see if it generates the sequence you want.
    It works thank you. Should I do something similar for the other questions?
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    (Original post by osayukiigbinoba)
    It works thank you. Should I do something similar for the other questions?
    Yep, try and see how.
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    (Original post by Zacken)
    Yep, try and see how.
    For 1,3,6,10,15 I got ur+1= ur+r+1
    But for 1,5,14,30,55 I have no idea. All I worked out so far was the difference 4,9,16,25 between each term. Then the second difference is 5,7,9.
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    (Original post by osayukiigbinoba)
    For 1,3,6,10,15 I got ur+1= ur+r+1
    Looks good!

    But for 1,5,14,30,55 I have no idea. All I worked out so far was the difference 4,9,16,25 between each term. Then the second difference is 5,7,9.
    What does the bolded bit remind you of...? Look at the first question for a reminder.

    Answer in spoiler:
    Spoiler:
    Show
    It's the sequence of squares! (shifted by a bit) So u_{r+1} = u_r + (r+1)^2
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    (Original post by Zacken)
    Looks good!



    What does the bolded bit remind you of...? Look at the first question for a reminder.

    Answer in spoiler:
    Spoiler:
    Show
    It's the sequence of squares! (shifted by a bit) So u_{r+1} = u_r + (r+1)^2
    Could you explain where (r+1)^2 comes from please?
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    (Original post by osayukiigbinoba)
    Could you explain where (r+1)^2 comes from please?
    You have:

    1 + 2^2 = 5
    5 + 3^2 = 14
    14 + 4^2 = ...

    In all cases you have:

    1st term: 1st term
    2nd term: 1st term + 2^2
    3rd term: 2nd term + 3^2
    4th term: 3rd term + 4^2
    (n+1)th term: (n)th term + (n+1)^2
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    (Original post by Zacken)
    You have:

    1 + 2^2 = 5
    5 + 3^2 = 14
    14 + 4^2 = ...

    In all cases you have:

    1st term: 1st term
    2nd term: 1st term + 2^2
    3rd term: 2nd term + 3^2
    4th term: 3rd term + 4^2
    (n+1)th term: (n)th term + (n+1)^2
    Thank you so much
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    (Original post by osayukiigbinoba)
    Thank you so much
    It really helps if you write all the sequences like I have above.
 
 
 
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