Question

Expand to 4 terms the following expressions: ${{\left( 1+\dfrac{1}{2}a \right)}^{-4}}$

Answer 1

Binomial expansion (or Binomial Theorem) which states that ${{\left( x+y \right)}^{n}}=\sum\limits_{r=0}^{n}{{}^{n}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}}$. Here use the binomial expansion for negative exponents i.e., $(1+x)^{-n} = 1 - nx + \dfrac{n(n+1)}{2!}x^2 + \dfrac{n(n+1)(n+2)}{3!}x^3 + . . . . .$

Complete step by step solution:

We have the expression ${{\left( 1+\dfrac{1}{2}a \right)}^{-4}}$. We have to write its expansion upto $4$ terms. We will use the formula for binomial expansion of terms which is $(1+x)^{-n} = 1 - nx + \dfrac{n(n+1)}{2!}x^2 + \dfrac{n(n+1)(n+2)}{3!}x^3 + . . . . .$

On substituting the values that is $n=-4$ and $x=\dfrac{1}{2}a$

${{\left( 1+\dfrac{1}{2}a \right)}^{-4}} = 1 - (-4)\left(\dfrac{1}{2}a\right) + \dfrac{(-4)(-4+1)}{2!}\left(\dfrac{1}{2}a\right)^2 + \dfrac{-4(-4+1)(-4+2)}{3!}\left(\dfrac{1}{2}a\right)^3$

On simplifying the above equation, we get

${{\left( 1+\dfrac{1}{2}a \right)}^{-4}} = 1 + (2a) + {(-2)(-3)}\left(\dfrac{1}{4}a^2\right) + \dfrac{-4(-3)(-2)}{3\times 2}\left(\dfrac{1}{8}a^3\right)$

${{\left( 1+\dfrac{1}{2}a \right)}^{-4}} = 1 + (2a) + \left(\dfrac{3}{2}a^2\right) - \left(\dfrac{1}{2}a^3\right)$

Hence we get the expansion of $\left(1+\dfrac{1}{2}a\right)^{4}$ upto 4 terms as $1+2a+\dfrac{3}{2}a^2-\dfrac{1}{2}a^3$

Note: Binomial expansion (also known as Binomial Theorem) describes the algebraic expansion of powers of a binomial. We expand the polynomial ${{\left( x+y \right)}^{n}}$ into a sum involving terms of the form $a{{x}^{b}}{{y}^{c}}$, where $b$ and $c$ are non-negative integers with $b+c=n$ and the coefficient $a$ of each term is a specific positive integer. The coefficient $a$ in the term $a{{x}^{b}}{{y}^{c}}$ is known as the binomial coefficient $\left( \begin{aligned} & n \\\ & b \\\ \end{aligned} \right)$ or $\left( \begin{aligned} & n \\\ & c \\\ \end{aligned} \right)$. These coefficients for varying $n$ and $b$ can be arranged to form a Pascal’s Triangle. While using the formula of binomial expansion, one must keep in mind that $n$ is a non-negative integer. That’s why to expand the expression ${{\left( 1+\dfrac{1}{2}a \right)}^{-4}}$, we wrote it in terms of fraction to get positive value of $n$.