Here's how to find the number Laverne says that sends Shirley running:

Shirley's Stopping Point:

We know Shirley runs away when the sum of the numbers Laverne says exceeds 1000.

Arithmetic Series:

The sequence of numbers Laverne says forms an arithmetic series with a common difference of 5 (since she counts by 5s). The first term is 2 (as she starts with 2).

Let n be the term number:

We want to find the value of n (the term number) at which the sum of the series becomes greater than 1000.

Formula for Arithmetic Series Sum:

The sum (S) of an arithmetic series can be calculated using the formula:

S = n/2 * (a1 + an)

where:

n = number of terms

a1 = first term

an = nth term (last term in this case)

Setting Up the Inequality:

We want the sum (S) to be greater than 1000. So, we can set up an inequality:

n/2 * (a1 + an) > 1000

Substituting Known Values:

a1 = 2 (first term)

an = a1 + (n-1) * d (nth term) where d is the common difference (5 in this case)

an = 2 + (n-1) * 5

Substitute these values into the inequality:

n/2 * (2 + (n-1) * 5) > 1000

Simplifying and Solving:

Expand the bracket:

n/2 * (2 + 5n - 5) > 1000

n/2 * (5n - 3) > 1000

Multiply both sides by 2 to eliminate the fraction:

n * (5n - 3) > 2000

Expand the left side:

5n² - 3n > 2000

Rearrange the inequality to isolate n:

5n² - 3n - 2000 > 0

Factoring the Expression:

This expression can be factored as:

(5n + 40) * (n - 50) > 0

Since the product of two factors is greater than zero only when both factors are positive or both are negative, we can consider two cases:

Case 1: (5n + 40) > 0 and (n - 50) > 0

Case 2: (5n + 40) < 0 and (n - 50) < 0

However, n (the term number) cannot be negative. So, we can disregard Case 2.

Solving Case 1:

From Case 1:

5n + 40 > 0

Subtract 40 from both sides: 5n > -40

Divide both sides by 5: n > -8

n - 50 > 0

Add 50 to both sides: n > 50

Since n represents the term number (positive integer), we are only interested in the positive solution that satisfies both inequalities.

The smallest positive integer value of n that fulfills both conditions is n = 51.

Finding the Laverne's Number:

Now that we know n (term number) is 51, we can find the corresponding number Laverne says that sends Shirley running.

Recall that the nth term (an) is calculated as:

an = a1 + (n-1) * d

an (51st term) = 2 + (51 - 1) * 5

an = 2 + 50 * 5

an = 252

Therefore, Laverne says 252 which makes the sum exceed 1000, sending Shirley running.