# Find the binomial expansion $(k+x)^n$

i) Find in terms of the non zero constant $k$, the first four terms on the expansion $(k+x)^n$ in ascending powers of $x$

ii) Given that the cooefficients of the $x^2$ and $x^{3}$ are equal, find the value of $k$.

I really neeed help on these so any help is highly appreciated!

No clue how to start. :(

• Do you really mean $x^{-3}$, or do you mean $x^3$? Because there is no term $x^{-3}$ in the expansion. – Thomas Andrews Jun 19 '13 at 19:23
• Oh sorry, your right it's x^3 – Sachin Jun 19 '13 at 19:23

HINT: At this point you’re expected to know the binomial expansion:

$$(k+x)^n=\sum_{i=0}^n\binom{n}ik^{n-i}x^i\;.$$

The first four terms are the terms for $i=0,1,2,3$:

$$\binom{n}0k^n,\quad\binom{n}1k^{n-1}x,\quad\binom{n}2k^{n-2}x^2,\quad\text{and}\quad\binom{n}3k^{n-3}x^3\;.$$

Since the coefficients of $x^2$ and $x^3$ are equal, you know that

$$\binom{n}2k^{n-2}=\binom{n}3k^{n-3}\;.$$

Divide out the common factors and expand the binomial coefficients into fairly simple fractions, and you’ll be able to solve for $k$ as a fairly simple function of $n$.

Hint: use binomial coefficients.