Rather than trying to memorise these specific formulae, I find it much easier to remember the basic, underlying concept is the same for both in discount factor/growth factor space:
\[ GF(0,T_2)=GF(0,T_1)*GF(T_1,T_2) \]
Then, depending on the question or data provided you just need to calculate the various discount factors appropriately. As David says above, the only difference between the two approaches is the compounding frequency and therefore discount (or growth) factor calculation.
As an aside, I should say that I find the presentation of the continuously compounded case in the screenshot confusing. Maybe I am just missing the associated background text to the numbers but my assumption would be that the quoted values would be the annual, continuously compounded rate. I therefore find the dividing by 2 and then multiply by 4 misleading. The general form is:
\[ GF=e^{rt} \]
where r is the annual continuously compounded rate and t is the period length in years. The full equation should therefore be:
\[ e^{0.02915*2}=e^{0.02136*1}*e^{r*1} \]
Obviously this works out the same but I don't see the rationale for the way it is presented in the text.
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