Here I have a question:
If the sum of the coefficients in the expansion of $(1+2x)^n $ is $ 6561$ then the greatest term in the expansion for $x=1/2 $ is?
So I used the formula for greatest term in an expansion and got:
$$ r < \frac {n+1} {2} $$
But I don't understand how to use the value of sum of coefficients in the problem. I don't know any formula for that. Also in the solutions I saw them taking $ x=1 $ and directly equating with $6561$ and getting $n=8$. What's the logic behind? I can't seem to understand.
Best Answer
Hint The sum $a_n + \cdots + a_0$ of the coefficients of a polynomial $p(x) := a_n x^n + \cdots + a_1 x + a_0$ coincides with $p(1)$.