Krisha, a heroin of Nepali film, made a music video and uploaded it to social media. In intervals of every hour, new viewers are increasing as given below:
| hour | 1 | 2 | 3 | ... |
| :----- | :---- | :---- | :---- | :-- |
| New viewers | 600 | 700 | 800 | ... |
a) In which sequence are new viewers of the video increasing?
b) How many total new viewers will there be till 9 hours? Find it.
c) Does the music make 8500 viewers in 9 hours? Calculate.
Answer (2)
The new viewers increase in an arithmetic sequence.
Calculate the total new viewers after 9 hours using the arithmetic series sum formula: S 9 = 2 9 ( 2 ( 600 ) + ( 9 − 1 ) ( 100 )) = 9000 .
Compare the total viewers (9000) with 8500.
The music video makes more than 8500 viewers in 9 hours, and the total number of viewers is 9000 .
Explanation
Understanding the Problem We are given the number of new viewers of a music video in the first few hours. We need to determine the type of sequence the new viewers are increasing in, the total number of new viewers till 9 hours, and whether the music video makes 8500 viewers in 9 hours.
Identifying the Sequence a) To determine the type of sequence, we check the difference between consecutive terms. The difference between the number of new viewers in the second hour and the first hour is 700 − 600 = 100 . The difference between the number of new viewers in the third hour and the second hour is 800 − 700 = 100 . Since the difference is constant, the sequence is an arithmetic progression.
Calculating Total Viewers b) To calculate the total number of new viewers after 9 hours, we use the formula for the sum of the first n terms of an arithmetic sequence: S n = 2 n ( 2 a 1 + ( n − 1 ) d ) , where n is the number of terms, a 1 is the first term, and d is the common difference. In this case, n = 9 , a 1 = 600 , and d = 100 . Substituting these values into the formula, we get: S 9 = 2 9 ( 2 ( 600 ) + ( 9 − 1 ) ( 100 )) = 2 9 ( 1200 + 800 ) = 2 9 ( 2000 ) = 9 × 1000 = 9000 So, the total number of new viewers after 9 hours is 9000.
Comparing with 8500 Viewers c) To determine if the music video makes 8500 viewers in 9 hours, we compare the total number of new viewers after 9 hours (which is 9000) with 8500. Since 8500"> 9000 > 8500 , the music video makes more than 8500 viewers in 9 hours.
Final Answer a) The new viewers are increasing in an arithmetic sequence. b) The total number of new viewers till 9 hours is 9000. c) Yes, the music video makes more than 8500 viewers in 9 hours.
Examples
Arithmetic sequences are useful in many real-life situations. For example, if you save a fixed amount of money each month, the total amount you've saved over time forms an arithmetic sequence. Similarly, if a machine depreciates in value by a fixed amount each year, the value of the machine over time forms an arithmetic sequence. Understanding arithmetic sequences can help you predict future values and make informed decisions in these scenarios.
The new viewers of Krisha's music video follow an arithmetic sequence. After 9 hours, the total number of new viewers is 9000, which is greater than 8500. Thus, the music video successfully makes more than 8500 viewers in that time frame.
;
