Evaluate the Summation sum from k=2 to 6 of (-1)^k(7-k)k
The problem involves evaluating a finite mathematical summation. The summation has bounds from k = 2 to 6, and the summand (the expression being summed) is given as (-1)^k(7-k)k. This expression must be calculated for each integer value of k from 2 to 6, with the results then being added together to reach the final sum.
In this summand:
- (-1)^k represents an alternating sign factor, meaning the sign changes with each increment of k (negative for odd k and positive for even k);
- (7-k) is a linear term which decreases as k increases;
- k is just the value of the summation index.
Each of these three components is multiplied together for each value of k in the given range, summed accordingly, to compute the total value of the summation.
$\sum_{k = 2}^{6} \left(\left(\right. - 1 \left.\right)\right)^{k} \left(\right. 7 - k \left.\right) k$
Answer
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:
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).
Exponents: The term $(-1)^k$ alternates the sign of the subsequent terms based on whether $k$ is even (positive sign) or odd (negative sign).
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.
Simplification: Each term of the expanded series is simplified by performing the arithmetic operations indicated.
Arithmetic Sum: The final step is to sum all the simplified terms to get the result of the summation.
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.
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.
