sum the following series to n Terms
 
2 + 10 + 30 + 68 + 130 +....
 
can we do two successive subtractions to get the nth term as an A.P. and then find the sum

Asked by ahuja8087 | 16th Nov, 2015, 07:08: PM

Expert Answer:

Given space series space is space 2 space plus space 10 space plus space 30 plus space 68 plus space 130 plus space...... equals left parenthesis 1 cubed plus 1 right parenthesis plus left parenthesis 2 cubed plus 2 right parenthesis plus left parenthesis 3 cubed plus 3 right parenthesis plus left parenthesis 4 cubed plus 4 right parenthesis plus left parenthesis 5 cubed plus 5 right parenthesis plus....... equals left parenthesis 1 cubed plus 2 cubed plus 3 cubed plus 4 cubed plus 5 cubed plus.... right parenthesis plus left parenthesis 1 plus 2 plus 3 plus 4 plus 5 plus........ right parenthesis  Let space straight T subscript straight n space be space the space nth space term space of space the space series space and space straight S subscript straight n space denotes space the space sum space of space its space straight n space terms.  straight T subscript straight n equals left parenthesis nth space term space of space 1 cubed plus 2 cubed plus 3 cubed plus 4 cubed plus 5 cubed plus.... right parenthesis plus left parenthesis nth space term space of space 1 plus 2 plus 3 plus 4 plus 5 plus........ right parenthesis straight T subscript straight n equals straight n cubed plus straight n  straight S subscript straight n equals sum from straight k equals 1 to straight n of straight T straight k equals sum from straight k equals 1 to straight n of open parentheses straight k cubed plus straight k close parentheses equals sum from straight k equals 1 to straight n of straight k cubed plus sum from straight k equals 1 to straight n of straight k equals open curly brackets fraction numerator straight n open parentheses straight n plus 1 close parentheses over denominator 2 end fraction close curly brackets squared plus fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction equals fraction numerator open curly brackets straight n left parenthesis straight n plus 1 right parenthesis close curly brackets squared plus 2 straight n left parenthesis straight n plus 1 right parenthesis over denominator 4 end fraction
 

Answered by Vijaykumar Wani | 17th Nov, 2015, 10:46: AM