{Use of Tech} Sum of squared integers Let T (n) = 1² + 2² + ... + n², where n is a positive integer. It can be shown that T (n) = n (n + 1) (2n + 1) / 8


c. What is the least value of n for which T(n) > 1000?

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the smallest positive integer n such that the sum of the squares of the first n integers, T(n), is greater than 1000. The formula for T(n) is given as T(n) = \( \frac{n(n + 1)(2n + 1)}{6} \).
2. Step 2: Set up the inequality. We want T(n) > 1000, so substitute the formula into the inequality: \( \frac{n(n + 1)(2n + 1)}{6} > 1000 \).
3. Step 3: Clear the fraction by multiplying both sides of the inequality by 6 to get: n(n + 1)(2n + 1) > 6000.
4. Step 4: Solve the inequality. This involves finding the smallest integer n that satisfies the inequality. You can start by testing integer values of n to see when the inequality holds true.
5. Step 5: Verify your solution. Once you find a candidate for n, substitute it back into the original formula for T(n) to ensure that T(n) > 1000.

Key Concepts

Sum of Squares Formula

The sum of the squares of the first n positive integers is given by the formula T(n) = n(n + 1)(2n + 1) / 6. This formula allows for the efficient calculation of the sum without needing to manually add each squared integer, which is particularly useful for large values of n.

Inequalities

An inequality is a mathematical statement that compares two expressions, indicating that one is greater than, less than, or not equal to the other. In this context, solving T(n) > 1000 requires finding the smallest integer n such that the sum of squares exceeds 1000, which involves manipulating the inequality to isolate n.

Integer Solutions

In many mathematical problems, especially those involving sums or sequences, we often seek integer solutions. For this question, we need to find the least positive integer n that satisfies the inequality, which may involve testing successive integer values or using numerical methods to find the threshold where T(n) first exceeds 1000.