Propyne is hydrogenated with $\mathrm{H}{}_2$ over Lindlar's catalyst to give P. P is treated with HBr in the presence of an organic peroxide to give Q. Q reacts with sodium acetylide, $\mathrm{HC}\equiv \mathrm{C}{}^{-}\mathrm{Na}{}^{+}$, to give R. R is

Lindlar's catalyst converts propyne into propene (P). Peroxide-catalysed HBr addition is anti-Markovnikov and gives 1-bromopropane (Q). The acetylide displaces bromide from this primary halide in an ${\mathrm{S}}_{\mathrm{N}}2$ reaction to give $\mathrm{HC}\equiv \mathrm{C}-\mathrm{CH}{}_2\mathrm{CH}{}_2\mathrm{CH}{}_3$, pent-1-yne. Hex-1-yne would need a four-carbon alkyl halide.