Krisha, a heroine of Nepali film, made a music video and uploaded it to social media. The number of new viewers is increasing every hour, as shown below:
| hour | 1 | 2 | 3 | ... |
|------------|-----|-----|-----|-----|
| New viewers | 600 | 700 | 800 | ... |
a) In which sequence are the new viewers of the video increasing?
b) How many total new viewers will there be after 9 hours? Find it.
c) Does the music video reach 8500 viewers in 9 hours? Calculate.
Answer (2)
The new viewers increase in an arithmetic sequence.
Calculate the number of viewers in the 9th hour: a 9 = 600 + ( 9 − 1 ) × 100 = 1400 .
Calculate the total number of viewers after 9 hours: S 9 = 2 9 ( 600 + 1400 ) = 9000 .
The music video did not make 8500 viewers in 9 hours. The final answer is 9000 .
Explanation
Understanding the Problem The problem provides the number of new viewers of a music video for the first three hours and asks us to identify the sequence, calculate the total viewers after 9 hours, and determine if the total viewers equal 8500. The number of new viewers increases each hour: Hour 1: 600 new viewers, Hour 2: 700 new viewers, Hour 3: 800 new viewers. The increase in viewers forms an arithmetic sequence.
Identifying the Sequence To determine the type of sequence, we find the difference between consecutive terms: 700 − 600 = 100 and 800 − 700 = 100 . Since the difference is constant, the sequence is an arithmetic progression with a common difference of 100.
Finding the Number of Viewers in the 9th Hour To calculate the total number of new viewers after 9 hours, we first need to find the number of new viewers in the 9th hour. We use the formula for the nth term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d , where a 1 is the first term, n is the number of terms (hours), and d is the common difference. In this case, a 1 = 600 , n = 9 , and d = 100 . Therefore, a 9 = 600 + ( 9 − 1 ) × 100 = 600 + 8 × 100 = 600 + 800 = 1400 .
Calculating the Total Number of Viewers After 9 Hours Now, we calculate the sum of the arithmetic series for the first 9 hours using the formula: S n = 2 n ( a 1 + a n ) , where S n is the sum of the first n terms, n is the number of terms (hours), a 1 is the first term, and a n is the nth term. In this case, n = 9 , a 1 = 600 , and a 9 = 1400 . Therefore, S 9 = 2 9 ( 600 + 1400 ) = 2 9 ( 2000 ) = 9 × 1000 = 9000 .
Comparing Total Viewers with 8500 Finally, we compare the total number of new viewers after 9 hours ( S 9 = 9000 ) with 8500. Since 9000 e q 8500 , the music video did not make 8500 viewers in 9 hours.
Final Answer a) The new viewers are increasing in an arithmetic sequence. b) The total number of new viewers after 9 hours is 9000. c) No, the music video did not make 8500 viewers in 9 hours.
Examples
Arithmetic sequences are not just abstract math; they appear in many real-world situations. For instance, consider a savings plan where you deposit a fixed amount each month. If you deposit $100 initially and then add $50 every month, the total amount you save each month forms an arithmetic sequence. Understanding arithmetic sequences helps you predict your savings over time, plan investments, and manage budgets effectively. This concept is also useful in calculating loan payments or understanding the depreciation of assets.
The new viewers of the music video are increasing in an arithmetic sequence. After 9 hours, there will be a total of 9000 new viewers, which exceeds 8500 viewers. Therefore, the music video does reach over 8500 viewers in that time frame.
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