# Algebra-Binomial Theorem

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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