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When a council offers free reflective house numbers, 30% of residents install them in the first month, the numbers in the second month are only 30% of those in the first month, and so on. What proportion of residents eventually installs them?

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Answer 1 · 2018-06-01T20:27:52

$42.8%$ of the residents.

This is a Geometric Progression that starts at $n = 1$ (instead of 0).

Let $R_0 =$ the initial number of residents.

Convert the percentage of residents that install the letters from $30%$ to a decimal:

$d = 0.3$

The number of residents that do the installation for any given month is:

$R_n = R_0d^n, {n in ZZ, n >=1}$

Let $S =$ the infinite sum

$S = R_0d+R_0d^2+R_0d^3+...$

From the reference, we know that:

$S = R_0+ R_0d+R_0d^2+R_0d^3+... = R_0/(1-d)$

To obtain the formula that does not start with $R_0$ , we subtract $R_0$ from the original formula:

$S = R_0d+R_0d^2+R_0d^3+... = R_0/(1-d) - R_0$

Substitute $0.3$ for $d$ :

$S = R_0/(1-0.3) - R_0$

Factor out R_0:

$S = R_0(1/(1-0.3) - 1)$

Use a calculator:

$S = R_0(0.bar(428571)...)$

This is approximately $42.8%$ of the residents.