3A. Compute the simplified numerical value of the expression: (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1).

The simplified numerical value of the expression: (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1) is 2021 and this can be determined by using the properties of arithmetic progression.

Given :

Expression -- (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1)

Let the given expression be written in the form of two arithmetic progressions. That is:

[tex]= \rm A_1-A_2[/tex]

where [tex]\rm A_1[/tex] is (4042 + 4040 + 4038 + … + 6 + 4 + 2) and [tex]\rm A_2[/tex] is (4041 + 4039 + 4037 + … + 5 + 3 + 1).

First, determine the number of terms in progression [tex]\rm A_1[/tex] and [tex]\rm A_2[/tex].

For progression [tex]\rm A_1[/tex] :

[tex]\rm 2 = 4042+(n-1)(-2)[/tex]

-4040 = -2(n - 1)

2020 = n - 1

2021 = n

For progression [tex]\rm A_2[/tex]:

[tex]1 = 4041+(n-1)(-2)[/tex]

-4040 = -2(n - 1)

2020 = n - 1

n = 2021

Sum of n terms for progression [tex]\rm A_1[/tex].

[tex]\rm S_n = \dfrac{2021}{2}\times (2(4042)+(2021-1)(-2))[/tex]

[tex]\rm S_n = 2021(4042-2020)[/tex]

[tex]\rm S_n = 2021\times 2022[/tex]

[tex]\rm S_n = 4086462[/tex]

Sum of n terms for progression [tex]\rm A_2[/tex].

[tex]\rm S'_n = \dfrac{2021}{2}\times (2(4041)+(2021-1)(-2))[/tex]

[tex]\rm S'_n=2021\times (4041-2020)[/tex]

[tex]\rm S'_n=2021\times (2021)[/tex]

[tex]\rm S'_n=4084441[/tex]

Now, the value of the expression [tex]\rm A_1-A_2[/tex] is:

[tex]= \rm A_1-A_2[/tex]

= 4086462 - 4084441

= 2021

So, the simplified numerical value of the expression: (4042 + 4040 + 4038 + … + 6 + 4 + 2) −(4041 + 4039 + 4037 + … + 5 + 3 + 1) is 2021.

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