Find the coefficient of x ^25 in the expansion of 1/1+ x + x^2 + x^3+ x^4

11+x+x2+x3+x4=11-x51-x=1-x1-x5We know 1-x-n=1+nx+nn-12!x2+nn-1n-23!x3+...+nn-1n-2...n-r-1r!xr+....11+x+x2+x3+x4=1-x1-x5=1-x1+x5+x10+x15+x20+x25+...=1+x5+x10+x15+x20+x25+...-x1+x5+x10+x15+x20+x25+...=1+x5+x10+x15+x20+x25+...-x+x6+x11+x16+x21+x26+...Notice the power of x in successive terms in each brackets is increasingTherefore only term of x25 comes from first bracketThus coefficient of x25 is 1

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