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 ________.
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 ________.
Correct Answer: 10620
Solution and Explanation
The correct answer is 10620
\(∵ |f(n)|≤800\)
\(⇒ -800 ≤ 2n^2-n-1≤800\)
\(⇒ 2n^2-n-801≤0\)
\(∴ n∈[\frac{-\sqrt{6409}+1}{4}, \frac{\sqrt{6409}+1}{4}]\) and \(n∈z\)
\(∴ n = -19, -18, -17,.....,19,20.\)
\(∴ ∑(2x^2-x-1) = 2∑x^2-∑x-∑1\)
\(= 2.2.(1^2+2^2+...+19^2)+2.20^2-20-40\)
= 10620
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Concepts Used:
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In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
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The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
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