The question is attached above.

Asked by sudhanshubhushanroy | 19th Nov, 2017, 09:14: PM

Expert Answer:

It is evident from question that each term in the sum begin mathsize 12px style stack sum I left parenthesis r right parenthesis with r equals 1 below and n on top end style
is perfect square. Our first aim is to get I(r).
 
begin mathsize 12px style R H S space o f space t h e space s u m space S space equals space 2 n cubed plus 9 n squared plus 13 n L e t space n space equals space 1 comma space S equals space 24 space equals space 6 cross times 4 space equals space 1 over 6 open parentheses 6 cross times 6 cross times 4 close parentheses space equals space 1 over 6 open parentheses 6 squared cross times 2 squared close parentheses L e t space n space equals space 2 comma space space S space equals space 78 space equals 24 plus 54 space equals space 24 plus space 1 over 6 open parentheses 6 squared cross times 3 squared close parentheses space equals space 1 over 6 open parentheses 6 squared cross times 2 squared close parentheses plus 1 over 6 open parentheses 6 squared cross times 3 squared close parentheses L e t space n space equals 3 comma space S space equals space 174 space equals space 78 plus 96 space equals 1 over 6 open parentheses 6 squared cross times 2 squared close parentheses plus 1 over 6 open parentheses 6 squared cross times 3 squared close parentheses space plus 1 over 6 open parentheses 6 squared cross times 4 squared close parentheses space H e n c e space sum from r equals 1 to n of I left parenthesis r right parenthesis space equals space sum from r equals 1 to n of fraction numerator 6 squared cross times open parentheses r plus 1 close parentheses squared over denominator 6 end fraction space space sum from r equals 1 to n of square root of I left parenthesis r right parenthesis end root space equals space sum from r equals 1 to n of fraction numerator 6 cross times left parenthesis r plus 1 right parenthesis over denominator square root of 6 end fraction space equals space square root of 6 space sum from r equals 1 to n of left parenthesis r plus 1 right parenthesis space equals square root of 6 space open square brackets 1 half n left parenthesis n plus 1 right parenthesis space plus space n close square brackets space equals space square root of 3 over 2 end root space space left parenthesis n squared plus 3 n right parenthesis    end style

Answered by  | 23rd Nov, 2017, 07:28: PM