Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 21 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?

Guest May 6, 2023

#1**0 **

*Pearl writes down seven consecutive integers, and adds them up. The sum of the integers is equal to 21 times the largest of the seven integers. What is the smallest integer that Pearl wrote down?*

Call the smallest integer x. Then each following integer

will be 1 larger than the integer that preceded it.

x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) + (x + 6) = 21 • (x + 6)

7x + 21 = 21x + 126

Combine like terms –14x = 105

Divide both sides by –14 x = –7**.**5

The problem fails already, since 7**.**5 is not

an integer, whether it's positive or negative. **NO SOLUTION**

But let's press ahead and see if this would have worked.

Check answer by substituting the value into the problem.

(–7**.**5) + (–7**.**5 + 1) + (–7**.**5 + 2) + (–7**.**5 + 3) + (–7**.**5 + 4) + (–7**.**5 + 5) + (–7**.**5 + 6) = (21)(–7**.**5 + 6)

(–7.5) + (–6**.**5) + (–5**.**5) + (–4**.**5) + (–3**.**5) + (–2**.**5) + (–1**.**5) = (21)(1**.**5)

–31**.**5 = –31**.**5

So, –7**.**5 would have been a good answer **IF** the problem

had not specified that the numbers have to be integers.

_{.}

Bosco Jun 30, 2023