Positive Integers: Evaluate $a+b+c$

In summary, positive integers are whole numbers greater than zero and can be evaluated by adding them together. The main difference between positive and negative integers is that positive integers are greater than zero while negative integers are less than zero. Any positive integers can be used in the expression $a+b+c$ and they can be applied in various real-life situations, such as counting, representing quantities, and mathematical operations.
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$a,\,b$ and $c$ are positive integers that satisfy the inequality $ab+3b+2c>a^2+b^2+c^2+3$. Evaluate $a+b+c$.
 
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We have$a^2 + b^2 + c^2 + 3 < ab + 3b +2c$or $(a^2-ab) + (b^2 - 3b) + (c^2 - 2c) + 3 < 0$or $(a-\frac{b}{2})^2 - \frac{b^2}{4} + (b^2 - 3b) + (c^2 - 2c) + 3< 0$Or $(a-\frac{b}{2})^2 + (3 \frac{b^2}{4} - 3b) + (c^2 - 2c) +3 < 0$Or $(a-\frac{b}{2})^2 + 3(\frac{b^2}{4} - b + 1) + (c^2 - 2c) < 0$Or $(a-\frac{b}{2})^2 + 3(\frac{b}{2} -1)^2 + (c^2 - 2c+1) < 1$Or $(a-\frac{b}{2})^2 + 3(\frac{b}{2} -1)^2 + (c-1)^2 < 1$For these to be true each of the terms has to be zero.(it can be seen that each of the terms which is squared if not zero has to be minimum $\frac{1}{4}$ giving sum 1 which is not true)So b = 2 a = 1 and c = 1 is the solution giving a+b+c = 4
 
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FAQ: Positive Integers: Evaluate $a+b+c$

1. What are positive integers?

Positive integers are whole numbers that are greater than zero. They can be written without any fractional or decimal components.

2. What is the meaning of "evaluate" in this context?

In mathematics, "evaluate" means to find the numerical value of an expression or equation. In the context of positive integers, it means to find the sum of three positive integers, represented by a, b, and c.

3. Can the positive integers be negative numbers?

No, by definition, positive integers cannot be negative numbers. They must be greater than zero to be considered positive integers.

4. How do you evaluate a+b+c if the values of a, b, and c are not given?

In order to evaluate a+b+c, you must have the specific values of a, b, and c. Without these values, the expression cannot be simplified or solved.

5. Can the sum of three positive integers be a negative number?

No, the sum of three positive integers will always be a positive number. This is because positive integers are, by definition, greater than zero.

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