why can we add accumulated values in example shown below?

Question

Can you explain to me why we're able to add the accumulated amount of 1000 invested for 5 years to the additional accumulated value of 1000 they invested after 3 years?

Answer 1

After $3$ years, there is

$$1000\cdot 1.01^{12}$$

in the account. Another $1000$ is deposited, and interest is applied for a further $2$ years:

$$\left(1000\cdot 1.01^{12}+1000\right)\cdot 1.01^{8}$$

This can be written as

$$1000\cdot 1.01^{12}\cdot 1.01^{8}+1000\cdot 1.01^{8}$$

$$=1000\cdot 1.01^{20}+1000\cdot 1.01^{8}$$

as in the solution.