+118 Laverne starts counting out loud by 5's. She starts with 2. As Laverne counts, Shirley sums the numbers Laverne says. When the sum finally exceeds 1000, Shirley runs screaming from the room. What number does Laverne say that sends Shirley screaming and running?
+279 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.
