A new movie is released each year for 10 years to go along with a popular book series. Each movie is 4 minutes longer than the last to go along with a plot twist. The first movie is 60 minutes long. Use an arithmetic series formula to determine the total length of all 10 movies.

Accepted Solution

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]Learn more:A problem to complete the square of quadratic function brainly.com/question/12992613A problem to determine the slope intercept form of a line brainly.com/question/1473992Inverse function brainly.com/question/1632445  Answer detailsGrade: Middle schoolSubject: 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.