The value of [tex]\rm S_{75}[/tex] is -675 and this can be determined by using the formula of the sum of n terms in the arithmetic sequence and the given data.
Given :
[tex]\rm a_n= 67-2n[/tex] --- (1)
The following steps can be used in order to determine the value of [tex]\rm S_{75}[/tex]:
Step 1 - Substitute the value of (n = 1) in the expression (1).
[tex]\rm a_1= 67-2(1)= 65[/tex]
Step 2 - Substitute the value of (n = 75) in the expression (1).
[tex]\rm a_{75}= 67-2(75)= -83[/tex]
Step 3 - Now, the formula of sum in an arithmetic sequence is given below:
[tex]\rm S_n=\dfrac{a_1+a_n}{2}\times n[/tex]
where n is the total number of terms, [tex]\rm a_1[/tex] is the first term, and [tex]\rm a_n[/tex] is the last term.
Step 4 - Substitute the values of the known terms in the above expression.
[tex]\rm S_{75}=\dfrac{65-83}{2}\times 75[/tex]
Step 5 - Simply the above expression.
[tex]\rm S_{75} = -675[/tex]
For more information, refer to the link given below:
https://brainly.com/question/10168678
