Generally speaking, if one pays an interest rate of $i_k$ in the $k$th month on a principal $P$ over $n$ months, then using the same derivation as that for monthly payments at a constant interest rate, I get the following general formula for the monthly payment $m$:
$$m = \frac{\displaystyle P \prod_{k=1}^n (1+i_k)}{\displaystyle 1+\sum_{\ell=2}^n \prod_{k=\ell}^n (1+i_k)}$$
In general, when only the first interest rate $i_1$ differs, the above formula may be vastly simplified using a geometric series in the denominator:
$$m = \frac{P i_2}{1-\left(1+i_2\right)^{-n}} \frac{1+i_1}{1+i_2}$$
In your case, $n=6$, $i_1=0.06$, $i_2=0.05$, and $P=1000$, so that $m \approx 198.89$. Note that this differs from the case in which all of the interest rates are at $5 \%$, in which the monthly payment becomes about $197.02$, by $1.87$ per month.