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.
