Permutations and Combinations

Question

Find the indicated terms in the following sequence whose nth terms are:

$straight h left parenthesis straight n right parenthesis space equals space fraction numerator straight n squared plus 2 straight n over denominator 2 straight n end fraction semicolon space straight h left parenthesis straight n minus 1 right parenthesis comma space straight h left parenthesis 16 right parenthesis$

Answer

Here,   $straight h left parenthesis straight n right parenthesis space equals space fraction numerator straight n squared plus 2 straight n over denominator 2 straight n end fraction$                                       ...(i)
Replacing n by n-1 in (1), we get
                      $straight h left parenthesis straight n minus 1 right parenthesis space equals space fraction numerator left parenthesis straight n minus 1 right parenthesis squared plus 2 left parenthesis straight n minus 1 right parenthesis over denominator 2 left parenthesis straight n minus 1 right parenthesis end fraction space equals space fraction numerator straight n squared minus 2 straight n plus 1 plus 2 straight n minus 2 over denominator 2 left parenthesis straight n minus 1 right parenthesis end fraction space equals fraction numerator straight n squared minus 1 over denominator 2 left parenthesis straight n minus 1 right parenthesis end fraction space equals space fraction numerator straight n plus 1 over denominator 2 end fraction$
Putting n = 16 in (I), we get $straight h left parenthesis 16 right parenthesis space equals space fraction numerator 16 squared plus 2 left parenthesis 16 right parenthesis over denominator 2 left parenthesis 16 right parenthesis end fraction space equals space fraction numerator 256 plus 32 over denominator 32 end fraction space equals space 288 over 32 space equals space 9$