I'm looking at Cherry section 1g #13

I know the answer to this kind of problem, but I'm not clear on the why...

given that a fund accumulates at a force of interest:

kt for 0<t<=5

.04kt^2 for 5<t<=10

and given that a deposit of 1 will accumulate to 2.7183 in 10 years, find k.

The problem I have is that I want to take the AV of the $1 after the first 5 years, then use that figure (here e^(25k/2)) as the deposit that accumulates for the next 5 years. So, I end up with e^(25k/2) times e^(875k/75). This is wrong, as it is necessary to add the two integrals together instead of mulitply.

Instinctually I know the correct way to proceed, but I get tripped up because I'm uncertain as to why I need to add instead of multiply.

Can anyone clear this up for me?

I know the answer to this kind of problem, but I'm not clear on the why...

given that a fund accumulates at a force of interest:

kt for 0<t<=5

.04kt^2 for 5<t<=10

and given that a deposit of 1 will accumulate to 2.7183 in 10 years, find k.

The problem I have is that I want to take the AV of the $1 after the first 5 years, then use that figure (here e^(25k/2)) as the deposit that accumulates for the next 5 years. So, I end up with e^(25k/2) times e^(875k/75). This is wrong, as it is necessary to add the two integrals together instead of mulitply.

Instinctually I know the correct way to proceed, but I get tripped up because I'm uncertain as to why I need to add instead of multiply.

Can anyone clear this up for me?

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