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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 29, 2023
 #1
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The answer is 15.

 Apr 29, 2023
 #2
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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 29, 2023
 #3
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n = 3.

How to obtain the smallest integer?

The seven integers are consecutive, hence they are represented in function of the smallest integer n as follows:

n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6.

The sum of the integers is equals to 14/3 times the largest integer, which is of (n + 6), hence the expression to obtain n is given as follows:

n + n + 1 + n + 2 + n + 3 + n + 4 + n + 5 + n + 6 = 14/3(n + 6)

7n + 21 = 14n/3 + 28

21n/3 - 14n/3 = 7

7n/3 = 7

7n = 21.

n = 3.

 May 1, 2023

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