Expand to 4 terms the following expressions: ${{\left( 1+\dfrac{1}{2}a \right)}^{-4}}$
Last updated date: 19th Mar 2023
•
Total views: 305.4k
•
Views today: 2.85k
Answer
305.4k+ views
Hint: 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:
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 \[\left( \begin{align}
& n \\
& b \\
\end{align} \right)\] or \[\left( \begin{align}
& n \\
& c \\
\end{align} \right)\]. 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 \[n\].
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 \[\left( \begin{align}
& n \\
& b \\
\end{align} \right)\] or \[\left( \begin{align}
& n \\
& c \\
\end{align} \right)\]. 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 \[n\].
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
