Find the sum upto nth term of the following series.
 
1) 1.3+2.5+3.7+........................
 

Asked by dkb.hap2010 | 22nd Sep, 2016, 01:33: PM

Expert Answer:

Given space series space is space 1.3 plus 2.5 plus 3.7 plus.......
nth space term space of space the space series space is space given space by comma
straight T subscript straight n equals straight n left parenthesis 2 straight n plus 1 right parenthesis equals 2 straight n squared plus straight n

Sum space of space first space straight n space terms space is space given space by comma
straight S equals space sum from straight k equals 1 to straight n of straight T subscript straight k equals sum from straight k equals 1 to straight n of open parentheses 2 straight n squared plus straight n close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space equals 2 sum from straight k equals 1 to straight n of straight n squared plus sum from straight k equals 1 to straight n of straight n
space space space space space space space space space space space space space space space space space space space space space space space space equals 2 open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 3 factorial end fraction close square brackets plus open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 factorial end fraction close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space equals 2 open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction close square brackets plus open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space equals straight n left parenthesis straight n plus 1 right parenthesis open square brackets fraction numerator 2 left parenthesis 2 straight n plus 1 right parenthesis plus 3 over denominator 6 end fraction close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space equals straight n left parenthesis straight n plus 1 right parenthesis open square brackets fraction numerator 4 straight n plus 5 over denominator 6 end fraction close square brackets
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 4 straight n plus 5 right parenthesis over denominator 6 end fraction

Answered by Vijaykumar Wani | 23rd Sep, 2016, 10:24: AM