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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?

 Jul 25, 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?  

 

Let the first integer be represented by x.   

Since the integers are consecutive, each following   

integer is one more than the integer before it.  

 

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

 

                                                             7x + 21  =  21x + 126    

 

                                                                 –14x  =  105     

 

                                                                       x  =  –7.5    

 

check answer    

 

(–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   

.

 Jul 25, 2023

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