If the total number of terms in the expansion of $(a-b+2c)^n=45$, then $n=?$

Question

Is there any specific formula for this because if there were two digits to expand then we can use the formula

$$\textrm{total number of terms} = n+1$$ But there are three digits to expand so i dont know how to do this Well this question has options and the options are

$$i) \ 7 \\ ii) \ 8 \\ iii) \ 9 \\ iv) \ 22$$

Answer 1

Consider a few cases: $$\begin{array}{c|c} n & N \\ \hline 0 & 1 \\ 1 & 3 \\ 2 & 6 \\ 3 & 10 \\ \vdots & \vdots \\ ? & 45 \end{array}$$ Note that $N$ is a triangular number. So: $$T_n=\frac{n(n+1)}{2}=45 \Rightarrow n=9.$$