Respuesta :
The sum of the length of all the ten movies is [tex]\fbox{\begin\\\ 780\text{ minutes}\\\end{minispace}}[/tex].
Step-by-step explanation:
It is given that a new movie is released each year for [tex]10[/tex] consecutive years so there are total number of [tex]10[/tex] movies released in [tex]10[/tex] years.
The movie released in first year is [tex]60\text{ minutes}[/tex] long and each movie released in the successive year is [tex]4\text{ minutes}[/tex] longer than the movie released in the last year.
So, as per the above statement movie released in first year is [tex]60[/tex] minutes long, movie released in second year is [tex]64[/tex] minutes long, movie released in third year is [tex]68[/tex] minutes long and so on.
The sequence of the length of the movie formed is as follows:
[tex]\fbox{\begin\\\ 60,64,68,72...\\\end{minispace}}[/tex]
The sequence formed above is an arithmetic sequence.
An arithmetic sequence is a sequence in which the difference between the each successive term and the previous term is always constant or fixed throughout the sequence.
The general term of an arithmetic sequence is given as
[tex]\fbox{\begin\\\math{a_{n} =a+(n-1)d}\\\end{minispace}}[/tex]
The sequence formed for the length of the movie is an arithmetic sequence in which the first term is [tex]60[/tex] and the common difference is [tex]4[/tex].
The arithmetic series corresponding to the arithmetic sequence of length of the movie is as follows:
[tex]\fbox{\begin\\\ 60+64+68+72+...\\\end{minispace}}[/tex]
The arithmetic series formula to obtain the sum of the above series is as follows:
[tex]\fbox{\begin\\\math{S_{n} =(n/2)(2a+(n-1)d)}\\\end{minispace}}[/tex]
In the above equation [tex]n[/tex] denotes the total number of terms, a denotes the first term, d denotes the common difference and Sn denotes the sum of n terms of the series.
Substitute [tex]\fbox{\begin\\\math{a}=60\\\end{minispace}}[/tex],[tex]\fbox{\begin\\\math{n}=10\\\end{minispace}}[/tex] and [tex]\fbox{\begin\\\math{d}=4\\\end{minispace}}[/tex] in the equation [tex]\fbox{\begin\\\math{S_{n} =(n/2)(2a+(n-1)d)}\\\end{minispace}}[/tex]
[tex]S_{10} =(10/2)(120+36) \\S_{10} =780[/tex]
Therefore, the length of the all [tex]10[/tex] movies as calculated above is [tex]\fbox{\begin\\\ 780\text{ minutes}\\\end{minispace}}[/tex]
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Answer details
Grade: Middle school
Subject: Mathematics
Chapter: Arithemetic preogression
Keywords: Sequence, series, arithmetic , arithmetic sequence, arithmetic series, common difference, sum of series, pattern, arithmetic pattern, progression, arithmetic progression, successive terms.
Answer:
The total length of all 10 movies is 780 minutes.
Further Explanation:
Arithmetic Sequence: A sequence of numbers in which difference of two successive numbers is constant.
The sum of n terms of an arithmetic sequence is given by the formula,
[tex]S_n=\dfrac{n}{2}[2a+(n-1)d][/tex]
Where,
- a is the first term of the sequence.
- d is a common difference.
- n is number of terms
- [tex]S_n[/tex] is sum of n terms of the sequence.
The first movie is 60 minutes long. This would be the first term of the sequence.
Thus, First term, a= 60 minutes
A new movie is released each year for 10 years. In 10 years total 10 movies will released.
Thus, Number of terms, n=10
Each movie is 4 minutes longer than the last released movie. It means the difference of length of two successive movie is 4 minutes.
Thus, Common difference, d=4
Using the sum of arithmetic sequence formula, the total length of all 10 movies is,
[tex]S_{10}=\dfrac{10}{2}[2\cdot 60+(10-1)\cdot 4][/tex]
[tex]S_{10}=\dfrac{10}{2}[2\cdot 60+9\cdot 4][/tex] [tex][\because 10-1=9][/tex]
[tex]S_{10}=\dfrac{10}{2}[120+36][/tex] [tex][\because 2\cdot 60=120\text{ and }9\cdot 4=36][/tex]
[tex]S_{10}=\dfrac{10}{2}\times 156[/tex] [tex][\because 120+36=156][/tex]
[tex]S_{10}=5\times 156[/tex] [tex][\because 10\div 2=5][/tex]
[tex]S_{10}=780[/tex] [tex][\because 5\times 156=780][/tex]
Therefore, The total length of all 10 movies is 780 minutes
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Keywords:
Arithmetic sequence, Arithmetic Series, Common difference, First term, AP progression, successive number, sum of natural number.
