If number of terms in the expansion of $(1 - 2/x + 4/x^2)^n$ is 29, then what is sum of coefficients of all the terms in this expansion ? To find the sum first of all we must find the value of n using given information which is really difficult because apparently both are functions of same varoable 2/x So how I can i solve this ? PS please use only the binomial theorem there is no need for multinomial theorem as there is only one variable x and I'm not very familiar with multinomial theorem.
How to find the sum of the coefficients in the expansion of a trinomial given nber of terms?
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linear-algebra
1 Answers
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Once we know $n$, the sum of coefficients is simply obtained by plugging in $x=1$ (why?), i.e., $3^n$.
The expansion will end with a multiple of $1/x^{2n}$ and of course start with $1$. Hence (unless some weird cancelling occurs that makes some coefficients zero) will have $2n+1$ terms. Hence 28 terms are not possible (I checked manually that no weird cancelling occurs for reasonable sizes of $n$). However, if the question instead wants to ask about terms up to $1/x^{28}$ (i.e., 29 terms), we may suppose that $n=14$.