Question 1209046: Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 56 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Before we dive into the problem, let's consider the task of adding the first n positive integers
1, 2, 3, ..., n-1, n
where n is a positive integer.
S = sum of those integers
S = 1+2+3+...+(n-1)+n
Make a copy of the sequence and reverse it so we have
| Original | 1 | 2 | 3 | ... | n-1 | n | | Reversed | n | n-1 | n-2 | ... | 2 | 1 |
Add straight down.
The resulting sum for each column is n+1.
There are n of these sums.
Adding n copies of (n+1) leads to n*(n+1)
This result is equal to 2*S because the sum of any entire row is S.
2*S = n*(n+1)
S = n*(n+1)/2
S = 0.5*n*(n+1)
An example:
Let's add the consecutive integers from 1 to 10.
0.5*n*(n+1) = 0.5*10*(10+1) = 55
You can check that 1+2+3+4+5+6+7+8+9+10 = 55
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Now we can get to the problem your teacher asked.
Let x be the smallest integer that Pearl thinks of.
Then x+1 is next integer, and x+2 is after that, and so on until reaching x+6 as the seventh consecutive integer.
Summing those items gives
x+(x+1)+(x+2)+(x+3)+(x+4)+(x+5)+(x+6)
= (x+x+x+x+x+x+x)+(1+2+3+4+5+6)
= 7x+(1+2+3+4+5+6)
= 7x+0.5*6*(6+1) ............... use the formula mentioned in the previous section
= 7x+21
We are then told that "The sum of the integers is equal to 56 times the largest of the seven integers."
So,
sum = 56*(largest value)
7x+21 = 56*(x+6)
This is where we run into a problem.
Solving this equation leads to x = -45/7 = -6.42857 approximately
The result is not an integer, so it seems like your teacher made a typo somewhere.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to
56 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?
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This problem is FATALLY DEFECTIVE, since it has no solution in integer numbers.
It describes a situation which is SELF-CONTRADICTORY and never may happen.
Good only to throw it to a garbage bin, where is its right place.
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