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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class11  >>  Sequence and Series
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If $1^4+2^4+3^4+.....n^4$ = $an^5+bn^4+cn^3+dn^2+en+f$, find $a$

$(a)\;1\qquad(b)\;\frac{1}{2}\qquad(c)\;\frac{1}{3}\qquad(d)\;\frac{1}{5}$

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1 Answer

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Answer : (d) $\frac{1}{5}$
Explanation : $1^4+2^4+3^4+\;----\;n^4=an^5+bn^4+cn^3dn^2+en+f$
$1^4+2^4+3^4+\;....\;(n+1)^4=a(n+1)^5+b(n+1)^4+c(n+1)^3+d(n+1)^2+e(n+1)+f$
Subtracting ,
$(n+1)^4=a\;[(n+1)^5-n^5]\;+b\;[(n+1)^4-n^4]\;+c\;[(n+1)^3-n^3]\;+d\;[(n+1)^2-n^2]\;+e\;[(n+1)-n]$
$5a=Coefficient\;of\;n^4\;in\;expansion\;of\;(n+1)^4$
$5a=1\quad\;a=\frac{1}{5}.$
answered Jan 20, 2014 by yamini.v
 

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