Respuesta :
The simplified numerical value of the given expression that is: (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1) is 2021 and this can be determined by using the formula of nth terms and sum of n terms in arithmetic progression.
Given :
Arithmetic Progression -- (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1)
The given expression can be written as the sum of two arithmetic progressions, that is:
[tex]\rm = B_1+B_2[/tex]
where [tex]\rm B_1[/tex] is (4042 + 4040 + 4038 + … + 6 + 4 + 2) and [tex]\rm B_2[/tex] is (4041 + 4039 + 4037 + … + 5 + 3 + 1).
The number of terms in progressions [tex]\rm B_1[/tex] and [tex]\rm B_2[/tex].
In progression [tex]\rm B_1[/tex] the total number of terms are:
[tex]2=4042+(n-1)(-2)[/tex]
-4040 = -2(n - 1)
2020 = n - 1
2021 = n
In progression [tex]\rm B_2[/tex] the total number of terms are:
[tex]1=4041+(n-1)(-2)[/tex]
-4040 = -2(n - 1)
2020 = n - 1
n = 2021
The sum of n terms in the arithmetic progression [tex]\rm B_1[/tex] is:
[tex]\rm S_n = \dfrac{2021}{2}(2(4042)+(2021-1)(-2))[/tex]
[tex]\rm S_n = {2021}(4042-2020)[/tex]
[tex]\rm S_n =4126882[/tex]
The sum of n terms in the arithmetic progression [tex]\rm B_2[/tex] is:
[tex]\rm S_n = \dfrac{2021}{2}(2(4041)+(2021-1)(-2))[/tex]
[tex]\rm S_n = {2021}(4041-2020)[/tex]
[tex]\rm S_n =4124861[/tex]
So, the value of [tex]\rm (B_1-B_2)[/tex] is:
= 4086462 - 4084441
= 2021
The simplified numerical value of the given expression that is: (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1) is 2021.
For more information, refer to the link given below
https://brainly.com/question/9230320
