For two consecutive first$-$order reactions:

A $->$ B $->$ C $($k$1,$ k$2)$

where CA$0=1$; CB$0=0$; CC$0=0$; k$1=0\mathrm{.}001$ s$-1$; k$2=0\mathrm{.}002$ s$-1$

Solve the rate expression equations and show that the solution for the concentration of C is given by:

CC $=$ CA$0\{1+[$k$2$exp$(-$k$1$t$)-$ k$1$exp$(-$k$2$t$)]/($k$1-$k$2)\}$

where CA$0,$ is the initial concentration of A$,$ and likewise CB$0$ and CC$0$ of B and C$,$ respectively.