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Question

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.