Evaluate the Summation sum from j=1 to 38 of j^3-25j

Solution:

Step 1:

Decompose the original summation into two separate summations using the distributive property of summation over subtraction.

$$\sum_{j = 1}^{38} (j^{3} - 25j) = \sum_{j = 1}^{38} j^{3} - 25 \sum_{j = 1}^{38} j$$

Step 2:

Calculate the summation of cubes up to the 38th term.

Step 2.1:

Use the cube summation formula:

$$\sum_{k = 1}^{n} k^{3} = \left(\frac{n(n + 1)}{2}\right)^2$$

Step 2.2:

Insert the upper limit of the summation into the formula.

$$\left(\frac{38(38 + 1)}{2}\right)^2$$

Step 2.3:

Perform the calculations.

Step 2.3.1:

First, work out the terms inside the brackets.

Step 2.3.1.1:

Sum $38$ and $1$.

$$\left(\frac{38 \cdot 39}{2}\right)^2$$

Step 2.3.1.2:

Square $38$.

$$\left(\frac{1444 \cdot 39}{2}\right)^2$$

Step 2.3.1.3:

Square $39$.

$$\left(\frac{1444 \cdot 1521}{2}\right)^2$$

Step 2.3.2:

Complete the square of the fraction.

Step 2.3.2.1:

Multiply $1444$ by $1521$.

$$\left(\frac{2196324}{2}\right)^2$$

Step 2.3.2.2:

Square the result.

$$549081$$

Step 3:

Compute the summation of the first 38 natural numbers multiplied by $-25$.

Step 3.1:

Apply the arithmetic series summation formula:

$$\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$$

Step 3.2:

Plug in the upper limit and include the multiplication by $-25$.

$$-25 \cdot \frac{38(38 + 1)}{2}$$

Step 3.3:

Simplify the expression.

Step 3.3.1:

Add $38$ and $1$.

$$-25 \cdot \frac{38 \cdot 39}{2}$$

Step 3.3.2:

Carry out the multiplication of $38$ and $39$.

$$-25 \cdot \frac{1482}{2}$$

Step 3.3.3:

Divide $1482$ by $2$.

$$-25 \cdot 741$$

Step 3.3.4:

Multiply $-25$ by $741$.

$$-18525$$

Step 4:

Combine the results from the two summations.

$$549081 - 18525$$

Step 5:

Finalize the subtraction to find the sum.

$$530556$$

Knowledge Notes:

The problem involves evaluating a summation of a polynomial expression. Here are the relevant knowledge points:

1. Summation Notation: The summation notation $\sum$ is used to denote the sum of a sequence of numbers. The expression under the summation sign indicates the pattern of the sequence, and the limits of the summation indicate the starting and ending indices.

2. Distributive Property: The summation operator can distribute over addition or subtraction. This property allows us to split a complex summation into simpler, more manageable parts.

3. Summation Formulas: There are known formulas for the summation of sequences of powers of integers, such as:

   • The sum of the first $n$ natural numbers: $\sum_{k = 1}^{n} k = \frac{n(n + 1)}{2}$
   • The sum of the cubes of the first $n$ natural numbers: $\sum_{k = 1}^{n} k^{3} = \left(\frac{n(n + 1)}{2}\right)^2$

4. Arithmetic Operations: Basic arithmetic operations (addition, subtraction, multiplication, division, exponentiation) are used to simplify expressions and calculate the final result.

5. Polynomial Expressions: A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

By applying these concepts, we can break down the original problem into simpler parts, use the appropriate summation formulas, perform arithmetic operations, and combine the results to find the final sum.