The first doll is 12 inches.
To get the next smallest doll (the second) we multiply 12 by 7/10.
To get the one after that (the third) we multiply 12 by 7/10 by 7/10
To get the one after that (the fourth) we multiply 12 by 7/10 by 7/10 by 7/10
and so on...
The height of each doll after the first will be multiplied by (7/10) raised to a power. Let's call n the number of the doll (first, second, third, ...). It might be easier to determine the power if we write it like this:
First Doll -> n =1 -> height = 12
Second Doll -> n=2 -> height = [tex](12)( \frac{7}{10}) ^{1} [/tex]
Third Doll -> n = 3 -> height = [tex](12)( \frac{7}{10}) ^{2} [/tex]
Fourth Doll -> n=4 -> height = [tex](12)( \frac{7}{10}) ^{3} [/tex]
Notice that the exponent of the fraction (7/10) is always one less than the value of n. This even works for the first doll. It's height is [tex](12)( \frac{7}{10}) ^{0} [/tex]
Thus, we have an expression for the height of the nth doll and it is [tex](12)( \frac{7}{10}) ^{n-1} [/tex]
The height of any doll can be found using this formula. Whatever the smallest doll is (the 10th, 27th, 99th, etc.) just use that number for n. Since we arrive at each height by multiplying the prior by (7/10) the heights form a geometric sequence.
