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23. For which positive integers n is the ratio, sum k = 1 to n k ^ 2 an integer? sum k = 1 k (A) odd n only (B) even n only
Asked by rareaditi4159 | 29 Sep, 2023, 08:25: PM
answered-by-expert Expert Answer
It is understood we have to find the ratio of begin mathsize 14px style space fraction numerator begin display style sum from k equals 1 to n of end style space k squared over denominator begin display style sum from k equals 1 to n of end style space k end fraction end style
 
begin mathsize 14px style sum from k equals 1 to n of k squared space equals space fraction numerator n space left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction end style
 
 
begin mathsize 14px style sum from k equals 1 to n of k space equals space fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style
 
begin mathsize 14px style space fraction numerator begin display style sum from k equals 1 to n of end style space k squared over denominator begin display style sum from k equals 1 to n of end style space k end fraction space equals space fraction numerator left parenthesis 2 n plus 1 right parenthesis over denominator 3 end fraction end style
 
above ratio is integer when , n = 1 , 4, 7 , 10 , 13 ................
Answered by Thiyagarajan K | 29 Sep, 2023, 09:06: PM

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23. For which positive integers n is the ratio, sum k = 1 to n k ^ 2 an integer? sum k = 1 k (A) odd n only (B) even n only
Asked by rareaditi4159 | 29 Sep, 2023, 08:25: PM
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JEE main - Maths
if an= -2/(4n^2)-(16n)+(15),      [(4n^2)-(16n)+(15) dis entire thing is the denominator)] then: opt a) a1+a2+....+a10= 20/51 opt b) a1+a2+.....a10=30/51 opt c) a1+a2+......a25=50/141 opt d) a1+a2+.....a25=60/141 Its more than one ans type q, pls tell me a detailed explanation on how to do it as i din understand 
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