Algebra-Binomial Theorem

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solution Expand (2x+3y)^{5}\,

solution Find the fourth term in ({\frac  {3}{x}}+y)^{7}\,

solution Find the middle term of the expansion ({\frac  {x}{y}}+{\frac  {y}{x}})^{8}\,

solution Find the middle terms of the expansion of (3x+{\frac  {1}{2x}})^{7}\,

solution Find the constant term in the expansion of (3x-{\frac  {5}{x^{2}}})^{9}\,

solution Find the independent term of x\, in the expansion of (x-{\frac  {1}{x}})^{6}\,

solution Find the independent term of x\, in the expansion of (6x^{2}-{\frac  {5}{x^{2}}})^{6}\,

solution Find the coefficient of x^{3}\, in the expansion of (3x+4)^{6}\,

solution Find the fifth term in the expansion of (2x+{\frac  {1}{3y}})^{8}\,

solution Prove that {C \choose 0}+{C \choose 2}+{C \choose 4}+...=2^{{n-1}}={C \choose 1}+{C \choose 3}+{C \choose 5}+.......\,

solution Prove that ({C \choose 0})^{2}+({C \choose 1})^{2}+({C \choose 2})^{2}+......+({C \choose n})^{2}={\frac  {(2n)!}{(n!)^{2}}}\,

solution Show that 676\, divides 3^{{3n}}-26n-1\,, where n\,is an integer.

solution If (1+x)^{n}={C \choose 0}+{C \choose 1}x+{C \choose 2}x^{2}+.......+{C \choose n}x^{n}\, then show that 1{C \choose 0}+2{C \choose 1}+3{C \choose 2}+........+(n+1){C \choose n}\,

solution Find the term containing x^{5}\, in the expansion of (x-{\frac  {1}{x}})^{{11}}\,

solution In the expansion of (1+a)^{n}\, if the coefficient of a^{{r-1}},a^{r},a^{{r+1}}\, are in AP.then prove that n^{2}-n(4r+1)+4r^{2}-2=0\,

solution Find the value of (0.99)^{9}\, corrected to 4 decimals using the binomial theorm.

solution Prove that {C \choose 1}^{{2}}-2{C \choose 2}^{{2}}+3{C \choose 3}^{{2}}-.....+2n{C \choose 2n}^{2}=(-1)^{{n-1}}n{C \choose 0}\,

solution Find the sum of the coefficients of the even powers of x in the expansion of (1+x+x^{2}+x^{3})^{5}\,

solution Prove that 2{C \choose 0}+2^{2}{\frac  {{C \choose 1}}{2}}+2^{3}{\frac  {{C \choose 2}}{3}}+......+2^{{n+1}}{\frac  {{C \choose n}}{n+1}}={\frac  {3^{{n+1}}-1}{n+1}}\,

solution Prove that {C \choose 0}+{\frac  {{C \choose 1}}{2}}x+{\frac  {{C \choose 2}}{3}}x^{2}+.....+{\frac  {{C \choose n}}{n+1}}x^{n}={\frac  {(1+x)^{{n+1}}-1}{(n+1)x}}\,


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