a) We showed in (b) that 7.9999... = 8

b) The sum of the geometric series that involves 7.9999... = 8

c) The number 8 has two decimal representations.

d) All real, rational numbers except for 0 have more than one decimal representations.

Step-by-step explanation:

x = 7.999999

To answer (a), we first evaluate (b)

b) We need to sum a geometric series to infinity to find the values of this expression.

7.99999 can be written as 7 + 0.9999

0.99999 is a geometric series that is essentially

0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 +

The sum to infinity for a geometric series is given as S = a ÷ (1-r)

where

a = first term = 0.9

r = common ratio = (second term) ÷ (first term) = 0.09 ÷ 0.9 = 0.1

Sum of this geometric series to infinity

= 0.9 ÷ (1 - 0.1) = 0.9 ÷ 0.9 = 1

0.9999... = 1

7.99999 = 7 + 0.99999... = 7 + 1 = 8

c) how many decimal representations does the number 8 have?

As shown in (b), 8 has 2 decimal representations, the one we know and this newly proven one.

d) which numbers have more than one decimal representation?

All real, rational numbers except for 0 have more than one decimal representations.

Hope this Helps!!!