The sum of all possible products taken two at a time out of the numbers ± 1, ± 2, ±3, ± 4 is
-30
Calculating the Sum of Products Taken Two at a Time
The problem asks us to find the sum of all possible products when we take two distinct numbers from the given set of numbers: \(\pm 1, \pm 2, \pm 3, \pm 4\).
The set of numbers is \(\{1, -1, 2, -2, 3, -3, 4, -4\}\). There are 8 distinct numbers in this set.
To find the sum of products taken two at a time from a set of numbers \(\{x_1, x_2, \dots, x_n\}\), we can use a helpful identity relating the square of the sum of the numbers to the sum of their squares and the sum of products taken two at a time.
The identity is:
\(\left( \sum_{i=1}^n x_i \right)^2 = \sum_{i=1}^n x_i^2 + 2 \sum_{1 \le i < j \le n} x_i x_j\)
Let \(S = \sum_{i=1}^n x_i\) be the sum of the numbers, and \(P_2 = \sum_{1 \le i < j \le n} x_i x_j\) be the sum of products taken two at a time. The identity can be written as:
\(S^2 = \sum x_i^2 + 2 P_2\)
We want to find \(P_2\), so we can rearrange the formula:
\(2 P_2 = S^2 - \sum x_i^2\)
\(P_2 = \frac{1}{2} \left( S^2 - \sum x_i^2 \right)\)
Step-by-Step Calculation
First, let's find the sum of the numbers (\(S\)) in the set \(\{1, -1, 2, -2, 3, -3, 4, -4\}\):
\(S = 1 + (-1) + 2 + (-2) + 3 + (-3) + 4 + (-4)\)
\(S = (1 - 1) + (2 - 2) + (3 - 3) + (4 - 4)\)
\(S = 0 + 0 + 0 + 0\)
\(S = 0\)
Next, let's find the sum of the squares of the numbers (\(\sum x_i^2\)):
The numbers are \(\pm 1, \pm 2, \pm 3, \pm 4\).
- Square of \(\pm 1\) is \(1^2 = (-1)^2 = 1\).
- Square of \(\pm 2\) is \(2^2 = (-2)^2 = 4\).
- Square of \(\pm 3\) is \(3^2 = (-3)^2 = 9\).
- Square of \(\pm 4\) is \(4^2 = (-4)^2 = 16\).
So, the sum of the squares is:
\(\sum x_i^2 = 1^2 + (-1)^2 + 2^2 + (-2)^2 + 3^2 + (-3)^2 + 4^2 + (-4)^2\)
\(\sum x_i^2 = 1 + 1 + 4 + 4 + 9 + 9 + 16 + 16\)
\(\sum x_i^2 = 2 \times (1 + 4 + 9 + 16)\)
\(\sum x_i^2 = 2 \times (30)\)
\(\sum x_i^2 = 60\)
Now, we can calculate the sum of products taken two at a time (\(P_2\)) using the formula \(P_2 = \frac{1}{2} \left( S^2 - \sum x_i^2 \right)\):
\(P_2 = \frac{1}{2} \left( (0)^2 - 60 \right)\)
\(P_2 = \frac{1}{2} \left( 0 - 60 \right)\)
\(P_2 = \frac{1}{2} \left( -60 \right)\)
\(P_2 = -30\)
Thus, the sum of all possible products taken two at a time out of the numbers \(\pm 1, \pm 2, \pm 3, \pm 4\) is -30.
Final Answer
The calculated sum of products taken two at a time is -30.
| Calculation Step | Result |
|---|---|
| Set of numbers | \(\{\pm 1, \pm 2, \pm 3, \pm 4\}\) |
| Sum of the numbers (\(S\)) | 0 |
| Sum of the squares (\(\sum x_i^2\)) | 60 |
| Sum of products two at a time (\(P_2\)) | \(-30\) |
Revision Table for Sum of Products
| Concept | Description | Formula |
|---|---|---|
| Sum of numbers | Adding all numbers in the set | \(S = \sum x_i\) |
| Sum of squares | Adding the squares of all numbers in the set | \(\sum x_i^2\) |
| Sum of products two at a time | Adding all unique products of pairs of numbers | \(P_2 = \sum_{i < j} x_i x_j\) |
| Relationship | Connects the sum, sum of squares, and sum of products two at a time | \((\sum x_i)^2 = \sum x_i^2 + 2 \sum_{i < j} x_i x_j\) |
Additional Information on Symmetric Polynomials
The sum of products taken two at a time is an example of an elementary symmetric polynomial. For a set of variables \(x_1, x_2, \dots, x_n\), the elementary symmetric polynomials \(e_k(x_1, \dots, x_n)\) are defined as:
- \(e_1 = \sum x_i\) (sum of the variables)
- \(e_2 = \sum_{i < j} x_i x_j\) (sum of products taken two at a time)
- \(e_3 = \sum_{i < j < k} x_i x_j x_k\) (sum of products taken three at a time)
- ...
- \(e_n = x_1 x_2 \dots x_n\) (product of all variables)
These polynomials are important in algebra, particularly in the study of roots of polynomials. For a polynomial \(P(t) = (t-x_1)(t-x_2)\dots(t-x_n)\), the coefficients are related to the elementary symmetric polynomials:
\(P(t) = t^n - e_1 t^{n-1} + e_2 t^{n-2} - e_3 t^{n-3} + \dots + (-1)^n e_n\)
In our problem, we calculated \(e_2\) for the set of numbers \(\{1, -1, 2, -2, 3, -3, 4, -4\}\). The relationship \((\sum x_i)^2 = \sum x_i^2 + 2 \sum_{i < j} x_i x_j\) is a specific case of Newton's sums or involving power sums and elementary symmetric polynomials.
The property that the sum of the numbers is zero (\(S=0\)) greatly simplified the calculation, as \(S^2\) became zero.
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