1.
Which represents the type of sequence:
33, 31, 28, 24, 19,...
option a.
option b.
option c.
option d.
2.
A town has a population of 32,000 in the year 2002; 38,400 in the year 2003; 46,080 in the year 2004; and 55,296 in the year 2005. If this pattern continues, what will the population be in the year 2015?
option a. 148,378
option b. 342,378
option c. 237,763
option d. 285,315
#1 c
#2
38400/32000 = 1.2
46080/38400 = 1.2
55296/46080 = 1.2
so, in 2015 it will be 55296* 1.2^9
1. The pattern in the sequence is decreasing by a certain amount each time.
To find the difference between consecutive terms, subtract the second term from the first term: 33 - 31 = 2.
To find the next term, subtract 2 from the previous term: 28 - 2 = 26.
Therefore, the next term in the sequence is 26.
This means that the answer is not among the given options.
2. To find the pattern in the population growth, calculate the ratio between consecutive terms.
The ratio between the population in 2003 and 2002 is 38,400 / 32,000 = 1.2.
The ratio between the population in 2004 and 2003 is 46,080 / 38,400 = 1.2.
The ratio between the population in 2005 and 2004 is 55,296 / 46,080 = 1.2.
Since the pattern continues with a constant ratio of 1.2, to find the population in 2015, divide the population in 2005 by the ratio 1.2 repeatedly.
55,296 * 1.2 = 66,355.2
66,355.2 * 1.2 = 79,626.24
79,626.24 * 1.2 = 95,551.48
95,551.48 * 1.2 = 114,661.78
114,661.78 * 1.2 = 137,594.14
137,594.14 * 1.2 = 165,112.97
165,112.97 * 1.2 = 198,135.57
198,135.57 * 1.2 = 237,762.69
Therefore, the population in the year 2015 would be approximately 237,763.
Hence, the answer is option c: 237,763.
1. To determine the type of sequence, we need to look for a pattern in the given numbers.
33, 31, 28, 24, 19,...
Looking at the differences between consecutive terms, we can see that:
33 - 31 = 2
31 - 28 = 3
28 - 24 = 4
24 - 19 = 5
The differences are increasing by 1 each time. This suggests that the sequence is following a decreasing pattern with a common difference of 1 in the differences. Therefore, the sequence is likely an arithmetic sequence.
Now, let's check the answer options to determine which corresponds to an arithmetic sequence:
a. If this option represents an arithmetic sequence, we would expect the differences between consecutive terms to be constant. Let's calculate the differences:
31 - 33 = -2
28 - 31 = -3
24 - 28 = -4
19 - 24 = -5
The differences are not constant, so option a is not the correct answer.
b. We can repeat the same process for option b, calculating the differences and checking their consistency. If the differences are not constant, option b is not the correct answer.
c. Repeating the process for option c, we find that the differences are not constant.
d. Finally, we repeat the process for option d:
33 - 31 = 2
31 - 29 = 2
29 - 27 = 2
27 - 25 = 2
The differences between consecutive terms are all equal to 2, which means the sequence is an arithmetic sequence with a common difference of 2. Therefore, option d represents the type of sequence in question.
2. To predict the population in the year 2015, we need to observe the growth pattern in the given years.
Year 2002: Population = 32,000
Year 2003: Population = 38,400
Year 2004: Population = 46,080
Year 2005: Population = 55,296
To find the growth factor between consecutive years, we can divide each population by the previous year's population:
Year 2003: 38,400 / 32,000 = 1.2
Year 2004: 46,080 / 38,400 = 1.2
Year 2005: 55,296 / 46,080 = 1.2
The growth factor between consecutive years is constant at 1.2. Therefore, we can use this growth factor to predict the population in the subsequent years.
To find the population in the year 2015, we need to determine the number of years from 2005 to 2015.
2015 - 2005 = 10 years
Now, we can calculate the predicted population:
Population in the year 2005 = 55,296
Predicted population in the year 2015 = Population in the year 2005 * (growth factor ^ number of years)
Predicted population in the year 2015 = 55,296 * (1.2 ^ 10)
Calculating this expression, we find the predicted population in the year 2015 is approximately 285,315.
Therefore, the correct answer is option d: 285,315.