Evaluate the Summation sum from k=2 to 4 of -k^2+k-3
The problem provided is a mathematical expression that requires the evaluation of a finite summation. Specifically, the summation involves a formula with a quadratic term (-k^2), a linear term (k), and a constant term (-3), and it needs to be computed for integer values of 'k' starting at 2 and ending at 4. The task is to find the cumulative result of this expression for each integer value of 'k' within the specified range and then sum those results to get the final answer.
$\sum_{k = 2}^{4} - k^{2} + k - 3$
Answer
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:
Expansion: We write out each term of the sequence by substituting the values of $k$ into the function $f(k)$.
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 $-29$.
