How many 3 letter words can be made from 4 letters “FGHI” if repetition is allowed and if repetition isn’t alllowed ?
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You can see how the difference between two consecutive terms is constantly increasing:
So, for the next terms we'll have to add +9, +11, +13 and so on.
Also, note that is obtained by adding the 2nd odd number to
,
is obtained by adding the 3rd odd number to
, and so on.
So, the recursive formula is
For the explicit formula, recall that the sum of the first n odd numbers is n squared. Taking into account the fact that we're not starting from 1, we have