i) Find the coeficient of 1/x⁵ in the expansion of (2x + 1/x)⁷
ii) Find the term which is independant of x in the expansion of (x² - 1/x)⁹
im in grade 11 and have an exam in 3 days. I tried to do them but get stuck or get a totally wrong answer!
i) Find the coeficient of 1/x⁵ in the expansion of (2x + 1/x)⁷
ii) Find the term which is independant of x in the expansion of (x² - 1/x)⁹
im in grade 11 and have an exam in 3 days. I tried to do them but get stuck or get a totally wrong answer!
Hi
Let's expand using Newton's binomial theorem : $\displaystyle \left(2x+\frac{1}{x}\right)^7=\sum_{k=0}^7\binom{7 }{k}(2x)^k\left(\frac{1}{x}\right)^{7-k}=\sum_{k=0}^7\binom{7}{k}2^k \underbrace{x^k}_{\frac{1}{x^{-k}}} \frac{1}{x^{7-k}}=\sum_{k=0}^7\binom{7}{k}2^k\frac{1}{x^{7-2k}}$
The coefficient of $\displaystyle \frac{1}{x^5}$ is $\displaystyle \binom{7}{k}2^k$ for $\displaystyle k$ such that $\displaystyle \frac{1}{x^{7-2k}}=\frac{1}{x^5} \Leftrightarrow 7-2k=5$