If the sum of coefficients in the expansion of (1+2x)^n is 6561 then the greatest binomial coefficient in the expansion is?

Question

I have managed to find out the general term as nC0+2*nC1+2^2nC2+....+2^n*nCn. How do I approach after this? Please help.

Answer 1

To find the sum of the coefficients of a polynomial, you just plug in $1$.

By plugging in $1$, we get $$3^n=6561, \quad n=\log_3 6561 = 8$$

Can you finish from here?

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The general term is now $$2^k \cdot _8C_k, k=0, 1, \cdots, 8$$ The maximum term occurs at $k=5, 6$ and its value is $1792$.

Answer 2

Hint:

The sum of coefficient in the expansion of $(1+2x)^n$ of which you have obtained the expression can be obtained by letting $x=1$.

That is $$3^n=6561$$