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# Need help

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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?

Apr 28, 2023

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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?

Since the integers are consecutive,

each integer's value will be one larger

than the integer before it  .

Call the first integer "x"

The seven integers are  (x) + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6)

and their sum is 7x + 21

We want this to be 21 times the largest number

so                                           7x + 21  =  21 times (x+6)

7x + 21  =  21x + 126

Collect like terms                    7x  – 21x  =  126 – 21

– 14x  =  105

x  =  –7.5

Check answer by plugging into the original proposition

(x)   +    (x+1)    +    (x+2)    +    (x+3)    +    (x+4)    +    (x+5)    +    (x+6)

(–7.5) + (–7.5+1) + (–7.5+2) + (–7.5+3) + (–7.5+4) + (–7.5+5) + (–7.5+6)

(–7.5)  +  (–6.5)   +   (–5.5)   +   (–4.5)   +   (–3.5)   +   (–2.5)   +   (–1.5)

Add all those together and the sum is       –31.5

Multiply 21 times –1.5 and the product is   –31.5       so the answer checks out good

.

Apr 28, 2023