Evaluate the Summation sum from k=2 to 6 of (-1)^k(7-k)k

Solution:

Step 1: Expansion of the Summation

Write out the summation for each individual term of $k$ from 2 to 6 using the given formula $(-1{)}^k(7-k)k$:

$(-1{)}^2(7-2)2+(-1{)}^3(7-3)3+(-1{)}^4(7-4)4+(-1{)}^5(7-5)5+(-1{)}^6(7-6)6$

Step 2: Simplification of Terms

Simplify each term of the expanded series:

$5\cdot 2-4\cdot 3+3\cdot 4-2\cdot 5+1\cdot 6$

Step 3: Calculation of the Sum

Add up all the simplified terms to find the final sum:

$10-12+12-10+6=6$

Solution:"6"

Knowledge Notes:

The problem involves evaluating a finite summation of a series where each term is defined by the function $(-1{)}^k(7-k)k$. Here's a breakdown of the relevant knowledge points:

1. Summation Notation: The summation notation $\sum$ is used to denote the addition of a sequence of numbers. The variable $k$ is the index of summation, which changes from the lower limit (in this case, 2) to the upper limit (in this case, 6).

2. Exponents: The term $(-1{)}^k$ alternates the sign of the subsequent terms based on whether $k$ is even (positive sign) or odd (negative sign).

3. Algebraic Expansion: This involves expanding the series by substituting the index $k$ with all the integers from the lower to the upper limit of the summation.

4. Simplification: Each term of the expanded series is simplified by performing the arithmetic operations indicated.

5. Arithmetic Sum: The final step is to sum all the simplified terms to get the result of the summation.

6. Even and Odd Functions: The function $(-1{)}^k$ is an example of an even-odd function, where the power of $-1$ determines the sign of the term. For even $k$, the term is positive, and for odd $k$, the term is negative.

7. Series and Sequences: A series is the sum of the terms of a sequence, which is a list of numbers following a certain pattern. In this problem, the sequence is defined by the function $(-1{)}^k(7-k)k$, and the series is the sum of this sequence for $k$ ranging from 2 to 6.