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    abbasi279's Avatar
    abbasi279 Posts: 1, Reputation: 1
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    #1

    Jan 4, 2011, 07:19 AM
    1,3,6... what is the nth term of the sequence?
    jcaron2's Avatar
    jcaron2 Posts: 986, Reputation: 204
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    #2

    Jan 4, 2011, 08:42 AM
    It looks like the first number was determined by adding 1 (to zero), the second number was determined by adding 2 to that, the third by adding an additional 3, etc.

    So to find the nth term, you need to add 1 + 2 + 3 +... + n.

    Luckily there's a shortcut for working out this sum.

    1 + 2 + 3 +... + n = (n/2)(n+1)

    If you want to know why, pretend you're trying to add the numbers 1 through 10.

    1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =??

    Try rearranging the order:

    1 + 10 + 2 + 9 + 3 + 8 + 4 + 7 + 5 + 6 =??

    Notice that each set of two numbers adds to 11:

    (1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) =??

    11 + 11 + 11 + 11 + 11 = 11 * 5 = 55

    So the ten numbers could be divided into 5 pairs (hence the n/2 in the above shortcut), each of which adds up to 11 (hence the n+1 in the shortcut). Likewise, if you wanted to add up the numbers from 1 to 100, you'd have 50 pairs of numbers adding up to 101. The (n/2)(n+1) shortcut works for any n, even if it's odd.

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