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# Algebra

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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 100, Shirley runs screaming from the room.  What number does Laverne say that sends Shirley screaming and running?

Apr 9, 2024

#2
+128794
+1

By  " brute force " .....

Laverne  says     2   7   12    17      22       27      32

Progressive sums     9      21    38        60       87      119

Another (more mathematical )  way

First term Laverne says = 2

Let the last term she says = 2 + 5n

Sum  > 100

[ 2 + (2 + 5n) ]  * n  / 2   > 100

[ 5n + 4 ] * n  >  200

5n^2 + 4n - 200 > 0      (1)            set =  0   and solve for positive n

5n^2 + 4n - 200 = 0

n = [ -4 + sqrt [ 16 + 4000] ] / 10    ≈  5.9

≈ 5.9 is a root of 5n^2 + 4n - 200....so....the next largest integer also  makes (1)  true

n = 6

And the number that Laverne says  is  2 + 5(6)  =    32

Apr 10, 2024
edited by CPhill  Apr 10, 2024

#1
+214
+1

She starts at 2, so she at least needs 98 more to go.

The only number that is divisible by 5 and is the smallest number greater than 98 is 100

So Laverne sends Shirley screaming and running at 2 + 100 = 102

Apr 10, 2024
#2
+128794
+1

By  " brute force " .....

Laverne  says     2   7   12    17      22       27      32

Progressive sums     9      21    38        60       87      119

Another (more mathematical )  way

First term Laverne says = 2

Let the last term she says = 2 + 5n

Sum  > 100

[ 2 + (2 + 5n) ]  * n  / 2   > 100

[ 5n + 4 ] * n  >  200

5n^2 + 4n - 200 > 0      (1)            set =  0   and solve for positive n

5n^2 + 4n - 200 = 0

n = [ -4 + sqrt [ 16 + 4000] ] / 10    ≈  5.9

≈ 5.9 is a root of 5n^2 + 4n - 200....so....the next largest integer also  makes (1)  true

n = 6

And the number that Laverne says  is  2 + 5(6)  =    32

CPhill Apr 10, 2024
edited by CPhill  Apr 10, 2024