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The sum of all possible products of the first n natural numbers taken two by two is ......explain by steps........ ..........second question is if nCr+nCr+1=n+1Cr then x is equal to ........ 

Asked by Farooq Ahmad 3rd May 2016, 6:35 PM
Answered by Expert
Answer:
begin mathsize 16px style W e space n e e d space t o space f i n d space t h e space s u m space o f space a l l space p o s s i b l e space p r o d u c t s space o f space t h e space f i r s t space apostrophe n apostrophe space n a t u r a l space n u m b e r s space t a k e n space t w o space b y space t w o.
T h a t space i s space w e space n e e d space t o space f i n d comma
S equals 1 cross times 2 plus 2 cross times 3 plus 3 cross times 4 plus 4 cross times 5 plus 5 cross times 6 plus....
space space space space equals sum from r equals 1 to n of n open parentheses n plus 1 close parentheses equals sum from r equals 1 to n of n squared plus sum from r equals 1 to n of n
W e space k n o w space t h a t comma
sum from r equals 1 to n of n equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction
sum from r equals 1 to n of n squared equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction
S u b s t i t u t i n g space t h e space a b o v e space v a l u e s space i n space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma
S equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction plus fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction
space space space space equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction cross times fraction numerator 2 n plus 1 over denominator 3 end fraction plus fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction
space space space space equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction cross times open square brackets fraction numerator 2 n plus 1 over denominator 3 end fraction plus 1 close square brackets
space space space space equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction cross times open square brackets fraction numerator 2 n plus 1 plus 3 over denominator 3 end fraction close square brackets
space space space space equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction cross times open square brackets fraction numerator 2 n plus 4 over denominator 3 end fraction close square brackets
space space space space equals fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction cross times open square brackets fraction numerator 2 open parentheses n plus 2 close parentheses over denominator 3 end fraction close square brackets
space space space space equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses n plus 2 close parentheses over denominator 3 end fraction
end style
Answered by Expert 4th May 2016, 10:02 AM
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