Answer:
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!!!
