Evaluate the Summation sum from k=2 to 4 of -k^2+k-3

Solution:

Step 1: Expand the Summation

Write out the terms of the summation for each integer value of $k$ starting at $k=2$ and ending at $k=4$:

$(-2^2+2-3)+(-3^2+3-3)+(-4^2+4-3)$

Step 2: Simplify the Expression

Step 2.1: Calculate $2^2$

Compute the square of $2$:

$-(2^2)+2-3+(-3^2)+3-3+(-4^2)+4-3$

Step 2.2: Multiply by $-1$

Apply the negative sign to the square of $2$:

$-4+2-3+(-3^2)+3-3+(-4^2)+4-3$

Step 2.3: Combine Terms

Add together $-4$ and $2$:

$-2-3+(-3^2)+3-3+(-4^2)+4-3$

Step 2.4: Continue Simplifying

Subtract $3$ from $-2$:

$-5+(-3^2)+3-3+(-4^2)+4-3$

Step 2.5: Calculate $3^2$

Compute the square of $3$:

$-5-(3^2)+3-3+(-4^2)+4-3$

Step 2.6: Multiply by $-1$

Apply the negative sign to the square of $3$:

$-5-9+3-3+(-4^2)+4-3$

Step 2.7: Combine Terms

Add together $-9$ and $3$:

$-5-6-3+(-4^2)+4-3$

Step 2.8: Continue Simplifying

Subtract $3$ from $-6$:

$-5-9+(-4^2)+4-3$

Step 2.9: Combine Terms

Subtract $9$ from $-5$:

$-14+(-4^2)+4-3$

Step 2.10: Calculate $4^2$

Compute the square of $4$:

$-14-(4^2)+4-3$

Step 2.11: Multiply by $-1$

Apply the negative sign to the square of $4$:

$-14-16+4-3$

Step 2.12: Combine Terms

Add together $-16$ and $4$:

$-14-12-3$

Step 2.13: Continue Simplifying

Subtract $3$ from $-12$:

$-14-15$

Step 2.14: Combine Terms

Subtract $15$ from $-14$ to get the final result:

$-29$

Knowledge Notes:

The problem involves evaluating a finite summation, which is a common operation in mathematics where you sum the values of a sequence of terms. The sequence is defined by a function of an integer variable that ranges over a set of consecutive integers. In this case, the function is $f(k)=-k^2+k-3$, and the variable $k$ ranges from $2$ to $4$.

To solve the problem, we follow these steps:

1. Expansion: We write out each term of the sequence by substituting the values of $k$ into the function $f(k)$.

2. Simplification: We perform arithmetic operations to simplify the expression. This includes:

   • Squaring the integers.

   • Multiplying by $-1$ to apply the negative sign.

   • Adding and subtracting terms to combine like terms.

The arithmetic operations used in the simplification process are based on the order of operations, also known as PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

The final result of the summation is a single integer, which is the sum of the evaluated terms of the sequence. In this case, the result is $-29$.