Answer
Verified
483.9k+ 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
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which places in India experience sunrise first and class 9 social science CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE