Find the sum of n terms of the series $open parentheses 4 minus 1 over straight n close parentheses space plus open parentheses 4 minus 2 over straight n close parentheses space plus open parentheses 4 minus 3 over straight n close parentheses space plus.........$

S space equals open parentheses 4 minus 1 over straight n close parentheses space plus open parentheses 4 minus 2 over straight n close parentheses space plus open parentheses 4 minus 3 over straight n close parentheses space plus......... space equals space left square bracket 4 space plus space 4 space plus space 4 space plus...... right square bracket space minus open square brackets 1 over straight n space plus 2 over straight n plus 3 over straight n plus.... close square brackets space equals space 4 left square bracket 1 plus 1 plus 1 plus.... right square bracket space minus 1 over straight n left square bracket 1 plus 2 plus 3 plus.... right square bracket space equals space straight S subscript 1 minus straight S subscript 2 straight S subscript 1 space equals space 4 space left square bracket 1 plus 1 plus 1 plus.... right square bracket straight a space equals 1 space comma straight d space equals 0 straight S subscript 1 space equals space 4 space straight x straight n over 2 left square bracket 2 space straight x 1 space plus left parenthesis straight n minus 1 right parenthesis space straight x space 0 right square bracket open parentheses straight S subscript straight n space equals space straight n over 2 left parenthesis 2 straight a space plus left parenthesis straight n minus 1 right parenthesis straight d right parenthesis close parentheses straight S subscript 1 space equals space 4 straight n straight S subscript 2 space equals space 1 over straight n left square bracket 1 plus 2 plus 3 plus...... right square bracket straight a space equals 1 comma space straight d equals 2 minus 1 space equals space 1 straight S subscript 2 space equals space 1 over straight n space straight x straight n over 2 left square bracket 2 space straight x 1 space plus left parenthesis straight n minus 1 right parenthesis space straight x 1 right square bracket space equals space 1 half left square bracket 2 plus straight n minus 1 right square bracket space equals space 1 half left square bracket 1 plus straight n right square bracket Thus comma space straight S space equals space straight S subscript 1 space minus straight S subscript 2 space equals space 4 straight n space minus 1 half space left square bracket 1 plus straight n right square bracket straight S space equals space fraction numerator 8 straight n minus 1 minus straight n over denominator 2 end fraction space equals space fraction numerator 7 straight n minus 1 over denominator 2 end fraction

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