Problem

Given that $k$ is a constant and the binomial expansion of

$$\sqrt{1+kx}\qquad |kx|<1$$

in ascending powers of $x$ up to the term in $x^3$ is

$$1+\frac{1}{8}x+Ax^2+Bx^3$$

(a) (i) find the value of $k$,

(ii) find the value of the constant $A$ and the constant $B$.

(b) Use the expansion to find an approximate value to $\sqrt{1.15}$

Show your working and give your answer to 6 decimal places.