If the number of integral terms in the expansion of (3^1/2+5^1/8)^n is exactly 33, then th…

Question

If the number of integral terms in the expansion of $\left(3^{1/2}+5^{1/8}\right)^n$ is exactly 33, then the least value of $n$ is:

Accepted Answer

$$T_{r+1}=\binom nr 3^{(n-r)/2}5^{r/8}$$

Both $\dfrac{n-r}2$ and $\dfrac r8$ must be integers, so $r$ must be a multiple of 8, and $n-r$ even (automatic since $r$ is a multiple of 8). For exactly 33 such terms ($r=0,8,16,\dots$, 33 values), the 33rd value is $r=8\times32=256$, requiring $n\ge256$.

Least value: $n=256$

Solutions · 1