Find the ##r^{th}## term of ##(a+2x)^n##

  • Thread starter Thread starter RChristenk
  • Start date Start date
  • Tags Tags
    Binomial theorem
AI Thread Summary
The discussion focuses on finding the ##r^{th}## term of the binomial expansion ##(a+2x)^n##, both from the beginning and the end. The ##r^{th}## term from the beginning is expressed as ##^nC_{r-1}a^{n-r+1}(2x)^{r-1}##, while the ##r^{th}## term from the end is given by ##^nC_{n-r+1}a^{r-1}(2x)^{n-r+1}##. Participants clarify the equivalence of the coefficients, noting that ##^nC_{r-1} = ^nC_{n-r+1}##, and discuss the correct formulation of these coefficients. The conversation also addresses the factorial representation of combinations and confirms the validity of the assumptions made regarding factorial expansions. Understanding these terms and their coefficients is crucial for accurately applying the binomial theorem.
RChristenk
Messages
73
Reaction score
9
Homework Statement
Find the ##r^{th}## term from the beginning and the ##r^{th}## term from the end of ##(a+2x)^n##
Relevant Equations
Binomial Theorem
##r^{th}## term from beginning: ##^nC_{r-1}a^{n-r+1}(2x)^{r-1}##

For the ##r^{th}## term from the end, we first know there are a total of ##n+1## terms in this binomial expansion. Subtracting the (##r^{th}## term from the end) from the total number of terms, ##n+1##, results in ##n+1-r## which represents the number of terms before it (counting from the beginning). Hence counting from the beginning, the ##r^{th}## term from the end is represented by ##n-r+2##.

Therefore ##r^{th}## term from the end: ##^nC_{n-r+1}a^{r-1}(2x)^{n-r+1}##.

The answers are given in an expanded coefficient form. For the ##r^{th}## term from the beginning: ##\dfrac{n(n-1)(n-2)...(n-r+2)}{(r-1)!}a^{n-r+1}(2x)^{r-1}##.

For the ##r^{th}## term from the end: ##\dfrac{n(n-1)(n-2)...(n-r+2)}{(r-1)!}a^{r-1}(2x)^{n-r+1}##.

At first, I didn't understand why the coefficient expansion was equivalent. Then I realized ##^nC_{r-1}=^nC_{n-r+1}##.

But technically ##^nC_{n-r+1} \neq \dfrac{n(n-1)(n-2)...(n-r+2)}{(r-1)!}##. Shouldn't it be something like ##\dfrac{n!}{(n-r+1)!}##? How would I write the full expansion of ##^nC_{n-r+1}##? Thanks.
 
Physics news on Phys.org
RChristenk said:
But technically ##^nC_{n-r+1} \neq \dfrac{n(n-1)(n-2)...(n-r+2)}{(r-1)!}##.
Why not?

We have <br /> {}^nC_k = \frac{n(n-1) \cdots (n - (k - 1))}{k!}. Setting k = n - r + 1 gives <br /> \begin{split}<br /> {}^nC_{n-r+1} &amp;= \frac{n(n-1) \cdots (n - (n - r + 1 - 1))}{(n - r + 1)!} \\<br /> &amp;= \frac{n(n-1) \cdots r}{(n - r + 1)!} \\<br /> &amp;= \frac{n!}{(r-1)!(n-r+1)!} \\<br /> &amp;= \frac{n(n-1) \cdots (n- r + 2)}{(r-1)!} \\<br /> &amp;= {}^nC_{r-1}.\end{split}

Shouldn't it be something like ##\dfrac{n!}{(n-r+1)!}##?

You are missing a factor of (r - 1)! from the denominator.
 
How did you go from ##\dfrac{n!}{(r-1)!(n-r+1)!}## to ##\dfrac{n(n-1)(n-2)\cdots(n-r+2)}{(r-1)!}##?

I am assuming ##n!=n(n-1)(n-2)\cdots (n-r+2)(n-r+1)(n-r)\cdots 1##. Would this be correct? Thanks.
 
RChristenk said:
How did you go from ##\dfrac{n!}{(r-1)!(n-r+1)!}## to ##\dfrac{n(n-1)(n-2)\cdots(n-r+2)}{(r-1)!}##?

I am assuming ##n!=n(n-1)(n-2)\cdots (n-r+2)(n-r+1)(n-r)\cdots 1##. Would this be correct? Thanks.
Your assumption is correct.
 
Back
Top