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Sir plz write step by step of the solution for easy understanding,thanks

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Asked by vishakhachandan026 18th June 2019, 4:41 PM
Answered by Expert
Answer:
S u m space o f space f i r s t space n space n a t u r a l space n u m b e r s equals fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction
S u m space o f space s q u a r e s space o f space f i r s t space n space n a t u r a l space n u m b e r s equals fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction
S u m space o f space c u b e s space o f space f i r s t space n space n a t u r a l space n u m b e r s equals fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction
rightwards double arrow S subscript 1 equals fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction comma space S subscript 2 equals fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction space space a n d space space S subscript 3 equals fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction
C o n s i d e r comma
fraction numerator S subscript 1 to the power of 4 space S subscript 2 squared minus S subscript 2 squared space S subscript 3 squared over denominator S subscript 1 squared plus S subscript 3 squared end fraction
equals fraction numerator S subscript 2 squared open parentheses S subscript 1 to the power of 4 space minus space S subscript 3 squared close parentheses over denominator S subscript 1 squared plus S subscript 3 squared end fraction
equals fraction numerator open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction end style close parentheses squared open square brackets open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses to the power of 4 minus open parentheses begin display style fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction end style close parentheses squared close square brackets over denominator open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses squared plus open parentheses begin display style fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction end style close parentheses squared end fraction
equals fraction numerator open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction end style close parentheses squared open square brackets begin display style fraction numerator n to the power of 4 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction end style minus begin display style fraction numerator n to the power of 4 left parenthesis n plus 1 right parenthesis to the power of 4 over denominator 16 end fraction end style close square brackets over denominator open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses squared plus open parentheses begin display style fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction end style close parentheses squared end fraction
equals fraction numerator open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction end style close parentheses squared open square brackets begin display style 0 end style close square brackets over denominator open parentheses begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close parentheses squared plus open parentheses begin display style fraction numerator n squared left parenthesis n plus 1 right parenthesis squared over denominator 4 end fraction end style close parentheses squared end fraction
equals 0
H e n c e comma space fraction numerator S subscript 1 to the power of 4 space S subscript 2 squared minus S subscript 2 squared space S subscript 3 squared over denominator S subscript 1 squared plus S subscript 3 squared end fraction equals 0.
Answered by Expert 18th June 2019, 6:20 PM
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