Question

Let
\(f(x) = 2x^2 - x - 1\ and\ S = \{ n \in \mathbb{Z} : |f(n)| \leq 800 \}\)
Then, the value of ∑_n∈S f(n) is equal to ________.

Answer 1

Correct Answer: 10620

To solve the problem, we need to find the set \(S = \{ n \in \mathbb{Z} : |f(n)| \leq 800 \}\) for the function \(f(x) = 2x^2 - x - 1\) and then compute \(\sum_{n \in S} f(n)\).

First, we solve \(|f(n)| \leq 800\) , i.e., \(-800 \leq 2n^2 - n - 1 \leq 800 \).

Consider \(2n^2 - n - 1 \leq 800\):

Rearrange:

\(2n^2 - n - 801 \leq 0\).

Find roots using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -1\), and \(c = -801\).

Calculate the discriminant:

\(D = (-1)^2 - 4 \cdot 2 \cdot (-801) = 1 + 6408 = 6409\).

Find roots:

\(n = \frac{1 \pm \sqrt{6409}}{4}\).

Calculate \(\sqrt{6409} = 80.06\) (approx.)

Roots:

\(n_1 = \frac{1 + 80.06}{4} \approx 20.265\), \(n_2 = \frac{1 - 80.06}{4} \approx -19.765\).

Since \(n\) is integer, valid \(n\) are: \(n \in [-19, 20]\).

Next, consider \(2n^2 - n - 1 \geq -800\):

Rearrange:

\(2n^2 - n + 799 \geq 0\).

Find roots:

\(D = (-1)^2 - 4 \cdot 2 \cdot 799 = 1 - 6392 = -6391\).

Since \(D\) is negative, \(2n^2 - n + 799 \geq 0\) holds for all \(n\).

Thus, \(n \in [-19, 20]\) satisfies both conditions.

Compute \(\sum_{n=-19}^{20} (2n^2-n-1)\).

Calculate:

\(\sum_{n=-19}^{20} 2n^2 = 2 \sum_{n=-19}^{20} n^2 = 2(5310)\), \(\sum_{n=-19}^{20} n = 0\), \(\sum_{n=-19}^{20} 1 = 40\).

Sum result:

\(2 \cdot 5310 - 0 - 40 = 10620\).

Therefore, the value is 10620, which fits the range (10620,10620).