Arithmetic Series and Product Calculation

This problem involves calculating the sum of an arithmetic series and then adding a product term.

Step 1: Sum of the Arithmetic Series

The series is $1 + 3 + 5 + 7 + \dots + 4033$. This is an arithmetic progression (AP) consisting of consecutive odd numbers.

• First term ($a$): 1
• Common difference ($d$): 2
• Last term ($l$): 4033

To find the sum, we first need the number of terms ($n$). Use the AP formula: $l = a + (n-1)d$.

$4033 = 1 + (n-1)2$

Subtract 1 from both sides:

$4032 = (n-1)2$

Divide by 2:

$2016 = n-1$

Solve for $n$:

$n = 2017$

Now, calculate the sum ($S_n$) using the formula $S_n = \frac{n}{2}(a+l)$.

$S_{2017} = \frac{2017}{2}(1 + 4033)$

$S_{2017} = \frac{2017}{2}(4034)$

$S_{2017} = 2017 \times 2017$

$S_{2017} = 2017^2$

Step 2: Calculate the Product

The second part of the expression is the product: $7983 \times 2017$.

Step 3: Combine the Sum and Product

The total value required is $(1+3+5+...+4033) + 7983 \times 2017$.

Substitute the sum calculated in Step 1:

$ 2017^2 + 7983 \times 2017 $

Notice that 2017 is a common factor. Factor it out:

$ 2017 \times (2017 + 7983) $

Calculate the sum inside the parentheses:

$ 2017 \times (10000) $

Perform the final multiplication:

$ 20170000 $

The final value is 20170000.