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Problem 35

$x$ is a natural number.

• The sum of all natural numbers from 1 to $x$ is denoted by $A$.
• The sum of all natural numbers from $x+1$ to 8 is denoted by $B$.
• The difference $A - B = 392$.

Find the value of $A$.

Solution:

1. Formula for the sum of natural numbers:
   $S = \frac{n(n+1)}{2}$
   Using this, $A$ can be written as: $A = \frac{x(x+1)}{2}$

2. Sum of numbers from $x+1$ to 8:
   The sum $B$ can be calculated as: B = \text{Sum from 1 to 8} - \text{Sum from 1 to x} = \frac{8(8+1)}{2} - \frac{x(x+1)}{2} Simplify: $B = 36 - \frac{x(x+1)}{2}$

3. Given $A - B = 392$:
   Substitute $A$ and $B$ into this equation: $\frac{x(x+1)}{2} - \left(36 - \frac{x(x+1)}{2}\right) = 392$ Simplify: $\frac{x(x+1)}{2} + \frac{x(x+1)}{2} - 36 = 392$ $x(x+1) - 36 = 392$ $x(x+1) = 428$

4. Solve for $x$:
   The equation $x(x+1) = 428$ is quadratic: $x^2 + x - 428 = 0$ Solve using the quadratic formula: $x = \frac{-1 \pm \sqrt{1 + 4(428)}}{2} = \frac{-1 \pm \sqrt{1713}}{2}$ Since $x$ must be an integer, test values near $\sqrt{1713} \approx 41$.

   Check $x = 20$: $20 \cdot 21 = 420 \quad (\text{Close!})$ Check $x = 21$: $21 \cdot 22 = 462 \quad (\text{Too large!})$

   Thus, $x = 20$, and: $A = \frac{20 \cdot 21}{2} = 210$

Answer for Problem 35: D) 210

Problem 36

$(2n + 2)$ and $(n - 1)$ are consecutive even integers. Find the product of all possible values of $n$.

Solution:

1. Express the relationship between consecutive integers:
   For $(2n + 2)$ and $(n - 1)$ to be consecutive even numbers: $2n + 2 = (n - 1) + 2$ Simplify: $2n + 2 = n + 1$ $n = -1$

   Thus, $n = -1$ is the only value possible.

2. Calculate the product of values:
   Since there is only one $n$, the product is: $\text{Product} = n = -1$

Answer for Problem 36: C) -1

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Related Questions:

1. How do you solve quadratic equations with non-integer roots?
2. Can you explain how to generalize the sum of numbers in different ranges?
3. What are some quick ways to check if two numbers are consecutive?
4. How do you interpret the difference between arithmetic sequences in such problems?
5. What happens if a quadratic equation has no real solutions?

Tip:

For problems involving the sum of integers, always write the sum formula clearly and verify boundary conditions to avoid errors.