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Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of $3\%$ for the first year, an effective rate of discount of $6\%$ for the second year, an effective rate of discount of 9% for the third year, and an effective rate of interest of $5\%$ for the fourth, fifth and sixth years?

A. $6.3\%\quad$ B. $6.4\%\quad$ C. $6.5\%\quad$ D. $6.6\%\quad$ E. $6.7\%\quad$

Answer for this Question is: The effective rate of (simple) interest would be: $$(1−0.03)^{-1}(1−0.06)^{-1}(1−0.09)^{-1}(1+0.05)^3=(1+6i)\implies i\approx 6.6\%$$

My question is how the first three values we are subtracting from 1 and last value is adding 1 why? Then for first three values they are putting power as $-1$ and last value they are putting power as $3$. Please how they solved and logic, please anyone guide me?

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According to Wikipedia, "the annual effective discount rate expresses the amount of interest paid/earned as a percentage of the balance at the end of the (annual) period". Thus if you have balance $p_{t+1}$ at the end of the period, and $p_t$ at the start of the period, then the effective discount rate, $d$ for that period is

$$d=\frac{p_{t+1}-p_t}{p_{t+1}}=1-\frac{p_t}{p_{t+1}}\iff p_{t+1}=\frac{p_t}{1-d}$$

whereas the effective interest rate, $r$ expresses the amount of interest as a percentage of the balance at the start of the period:

$$r=\frac{p_{t+1}-p_t}{p_{t}}=\frac{p_{t+1}}{p_t}-1\iff p_{t+1}=(1+r)p_t.$$

Thus in your problem, a dollar at the start of the period of 6 years will become $A$ dollars at the end, where

$$A=\left(\frac{1}{1-0.03}\right)\left(\frac{1}{1-0.06}\right)\left(\frac{1}{1-0.09}\right)(1+0.05)^3 $$

The simple rate of interest $i$ equivalent to this must satisfy

$$1+6i=A\iff i=\frac{A-1}{6}$$

Doing the calculation gives $i\approx 0.66$.

Note that we can easily show that the effective discount rate, $d$ and effective interest rate $r$ are related as follows:

$$d=\frac{r}{1+r}\iff r=\frac{d}{1-d}$$

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