So I have derived the formula for the numerically greatest term of the following expression
$$(a+b)^n$$
It is given by taking $T_r$ and $T_{r+1}$ and using the fact that
$$\frac{T_{r+1}}{T_r} \geq 1$$ (or reciprocal depending on "which" r we need)
where $T_r$ is the r'th term of the binomial expansion of $(a+b)^n$ and r goes from $1$ to $(n+1)$
So basically $T_r = \hspace{0.1cm} ^nC_r \hspace{0.1cm}a^{n-r}\hspace{0.1cm}b^r$ in the expansion of $(a+b)^n$
Important : Note that we are supposed to expand $(a+b)^n$ first and then put in values of a and b
So finally the formula is : $$r\geq \frac{1+n}{1+ |\frac{a}{b}|}$$
we then simply find the $T_r$ which would be the NGT. Done !
My confusion is which to take as 'a' and which as 'b'. Wouldn't they be interchangeable ?
Or maybe ... does the result apply for both 'a' and 'b' (any order) ?
(But no it doesn't !!)
Now what is the correct formula ?
for a quick example refer this Q/A :
[1] What is the numerically greatest or magnitude wise greatest term in the expansion?
[2] Greatest term in the expansion $(2+3x)^9$
Okay . . . revisiting my question again after going through the comments, here is the question which I would like answered:
I understood that my formula was wrong. We are not supposed to open up $(a+b)^n$ just as it is. We are required to factor out either a or b after which we end up with the binomial of $(1+x)^n$. Then again applying $T_{r+1} \geq T_{r}$ we get $$r \geq \frac{(1+n)|x|}{(1+|x|)}$$
Here's the issue : Which exactly do i factor out: a or b?