$\text{ If the }{10}^{\text{th }}\text{ term of H.P is }21\text{ and }{21}^{\text{st }}\text{ term of H.P is }10\text{ then }{420}^{\text{th }}\text{ term is }$

${10}^{\text{th }}\text{ terms of }\mathrm{H}.\mathrm{P}=\frac{1}{a+9d}=21,{\mathrm{t}}_{21}=\frac{1}{a+20d}=10$

$\begin{array}{l}\to \mathrm{a}+9\mathrm{d}=\frac{1}{21}--\left(1\right)\\ \mathrm{and} \mathrm{a}+20\mathrm{d}=\frac{1}{10}--\left(2\right)\\ \left(2\right)-\left(1\right)=11\mathrm{d}=\frac{1}{10}-\frac{1}{21}=\frac{11}{210}\\ \Rightarrow \mathrm{d}=\frac{1}{210}and a=\frac{1}{210}\\ {\mathrm{T}}_{420}=\frac{1}{\frac{1}{210}+\frac{419}{210}}=\frac{1}{2}\end{array}$