Find the coefficient of ${x^n}$ in the expansion of ${e^{a - bx}}$.
Last updated date: 21st Mar 2023
•
Total views: 305.1k
•
Views today: 6.83k
Answer
305.1k+ views
Hint: ${e^{a - bx}}$ can be written as ${e^a}.{e^{ - bx}}$. And then the expansion of ${e^x}$ is $1 + \dfrac{x}{{1!}} + \dfrac{{{x^2}}}{{2!}} + ....$
Use this expansion for ${e^{ - bx}}$ and find the general term when it is multiplied by ${e^a}$.
Complete step-by-step answer:
According to the question, we have to find out the coefficient of ${x^n}$ in the expansion of ${e^{a - bx}}$.
We can write ${e^{a - bx}}$ as:
$ \Rightarrow {e^{a - bx}} = {e^a}.{e^{ - bx}} .....(i)$
We know the expansion of ${e^x}$ is:
$ \Rightarrow {e^x} = 1 + \dfrac{x}{{1!}} + \dfrac{{{x^2}}}{{2!}} + ....$
Using this expansion for ${e^{ - bx}}$ in equation $(i)$, we’ll get:
\[ \Rightarrow {e^{a - bx}} = {e^a} \times \left[ {1 + \dfrac{{\left( { - bx} \right)}}{{1!}} + \dfrac{{{{\left( { - bx} \right)}^2}}}{{2!}} + \dfrac{{{{\left( { - bx} \right)}^3}}}{{3!}}...\dfrac{{{{\left( { - bx} \right)}^n}}}{{n!}}...} \right]\]
In the above expansion, ${x^n}$ will occur for the term ${e^a}.\dfrac{{{{\left( { - bx} \right)}^n}}}{{n!}}$.
This term can be written as ${e^a}.\dfrac{{{{\left( { - b} \right)}^n}}}{{n!}}{x^n}$.
Thus, the coefficient of ${x^n}$ in the expansion of ${e^{a - bx}}$ is ${e^a}.\dfrac{{{{\left( { - b} \right)}^n}}}{{n!}}$.
Note: We could have used the expansion of ${e^{a - bx}}$ directly as:
$ \Rightarrow {e^{a - bx}} = 1 + \dfrac{{\left( {a - bx} \right)}}{{1!}} + \dfrac{{{{\left( {a - bx} \right)}^2}}}{{2!}} + \dfrac{{{{\left( {a - bx} \right)}^3}}}{{3!}} + ....$
Although theoretically we will get the same result as above but it’s not possible to find it out in this case because every term will contain a mixture of different powers of $x$ and the expansion is also going up to infinity.
Use this expansion for ${e^{ - bx}}$ and find the general term when it is multiplied by ${e^a}$.
Complete step-by-step answer:
According to the question, we have to find out the coefficient of ${x^n}$ in the expansion of ${e^{a - bx}}$.
We can write ${e^{a - bx}}$ as:
$ \Rightarrow {e^{a - bx}} = {e^a}.{e^{ - bx}} .....(i)$
We know the expansion of ${e^x}$ is:
$ \Rightarrow {e^x} = 1 + \dfrac{x}{{1!}} + \dfrac{{{x^2}}}{{2!}} + ....$
Using this expansion for ${e^{ - bx}}$ in equation $(i)$, we’ll get:
\[ \Rightarrow {e^{a - bx}} = {e^a} \times \left[ {1 + \dfrac{{\left( { - bx} \right)}}{{1!}} + \dfrac{{{{\left( { - bx} \right)}^2}}}{{2!}} + \dfrac{{{{\left( { - bx} \right)}^3}}}{{3!}}...\dfrac{{{{\left( { - bx} \right)}^n}}}{{n!}}...} \right]\]
In the above expansion, ${x^n}$ will occur for the term ${e^a}.\dfrac{{{{\left( { - bx} \right)}^n}}}{{n!}}$.
This term can be written as ${e^a}.\dfrac{{{{\left( { - b} \right)}^n}}}{{n!}}{x^n}$.
Thus, the coefficient of ${x^n}$ in the expansion of ${e^{a - bx}}$ is ${e^a}.\dfrac{{{{\left( { - b} \right)}^n}}}{{n!}}$.
Note: We could have used the expansion of ${e^{a - bx}}$ directly as:
$ \Rightarrow {e^{a - bx}} = 1 + \dfrac{{\left( {a - bx} \right)}}{{1!}} + \dfrac{{{{\left( {a - bx} \right)}^2}}}{{2!}} + \dfrac{{{{\left( {a - bx} \right)}^3}}}{{3!}} + ....$
Although theoretically we will get the same result as above but it’s not possible to find it out in this case because every term will contain a mixture of different powers of $x$ and the expansion is also going up to infinity.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

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

Tropic of Cancer passes through how many states? Name them.

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

Name the Largest and the Smallest Cell in the Human Body ?
