Answer:
The 75th term of the arithmetic sequence -17, -13, -9.... is:
[tex]a_{75}=279[/tex]
Step-by-step explanation:
Given the sequence
[tex]-17, -13, -9....[/tex]
An arithmetic sequence has a constant difference 'd' and is defined by
[tex]a_n=a_1+\left(n-1\right)d[/tex]
computing the differences of all the adjacent terms
[tex]-13-\left(-17\right)=4,\:\quad \:-9-\left(-13\right)=4[/tex]
The difference between all the adjacent terms is the same and equal to
[tex]d=4[/tex]
The first element of the sequence is:
[tex]a_1=-17[/tex]
now substitute [tex]d=4[/tex] and [tex]a_1=-17[/tex] in the nth term of the sequence
[tex]a_n=a_1+\left(n-1\right)d[/tex]
[tex]a_n=4\left(n-1\right)-17[/tex]
[tex]a_n=4n-21[/tex]
Now, substitute n = 75 in the [tex]a_n=4n-21[/tex] sequence to determine the 75th sequence
[tex]a_n=4n-21[/tex]
[tex]a_{75}=4\left(75\right)-21[/tex]
[tex]a_{75}=300-21[/tex]
[tex]a_{75}=279[/tex]
Therefore, the 75th term of the arithmetic sequence -17, -13, -9.... is:
[tex]a_{75}=279[/tex]
