1. If n is a positive integer and the coefficient of x^2 in the expansion of (1+x)^n+(1+2x)^n is 75, find the value of n.
2. Expand [(1+x)^n][(1-2x)^4] in ascending powers of x up to the term x^2, where n is a positive integer. If the coefficient of x^2 is 54, find the coefficient of x.
3. Solve the equation ln(1+x)=1+lnx, giving your answer correct to 2 sig fig.
4a. Expand (1+2x)^n in ascending powers of x up to the term x^3, where n is a positive integer.
4b. In the expansion of [(x-3/x)^2][(1+2x)^n], the constant term is 210. Find the value of n.
5.Given that the expansion of (a+x)(1-2x)^n in ascending powers of x is 3-41x+bx^2+..., find the values of the constants a, n and b.
6a. In the binomial expansion of (x+k/x^3)^8, where k is a positive constant, the term independent of x is 252. Evaluate k.
6b. Using your value of k in question 6a, find the coefficient of x^4 in the expansion of (1-x^4/4)(x+k/x^3)^8.
7. Solve the equation (3^x+1)-2=8(3^x-1).
更新1:
RE:超凡學生 In question 4a, the answer should be expanded to x^3 but not x^2.
