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# help~!

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7.  The numbers 19, a, b, c, 73 form an arithmetic sequence, in that order.
Find a+b+c.

8. The number 8!+9!+10! is equal to n*8! for some integer n. What
is the value of n?

9. Regular octagon ABCDEFGH has side length 4 units. What is the

10. The least common multiple of 210 and n is 210n, where n is an integer
greater than 1. What is the smallest possible value of n?

11. The side lengths of an acute triangle, in inches, are integers and form
an increasing arithmetic sequence. What is the smallest possible value
for the perimeter of the triangle?

Feb 18, 2018

#1
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7)

73 -19 =54

54 / 4 = 13.5 - common difference

19, 32.5, 46, 59.5, 73.........etc.

a + b + c = 32.5 + 46 + 59.5 = 138

8)

[8! + 9! + 10!] =8! x [(1 + 9 + 90)] / 8! x n

n = 100

Feb 18, 2018
edited by Guest  Feb 18, 2018
edited by Guest  Feb 18, 2018
edited by Guest  Feb 18, 2018
edited by Guest  Feb 18, 2018
#5
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Yes, I think both of them are correct......

tertre  Feb 18, 2018
#2
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How did you get 54/3=13.5?

Feb 18, 2018
edited by azsun  Feb 18, 2018
edited by azsun  Feb 18, 2018
#3
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Typo !!!!!!!!!?.

Feb 18, 2018
#4
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Wait till CPhill comes. He's great at this.

Feb 18, 2018
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9.  Interior angles of a regular polyhedron total   (n-2)(180)

for an octagon    (8-2)(180) = 1080    there are 8 interior angles so each is 1080/8 = 135 degrees

The sides of the SQUARE form isocoles triangles of sides 4, 4   and 'x' (the side of the square) with an included 135 degree angle beteen the 4 and 4 sides

The other two angles are   (180-135)/2 = 22.5 degrees

Use the law of sines to find 'x'

sin 135 /x  = sin22.5  / 4    results in square side length of 7.39

Square Area = 7.39 x 7.39 =54.63 units^2

Feb 18, 2018
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11.     The smallest I can find ...by trial and error    is    4  5   6

perimeter = 15

Feb 18, 2018
#8
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Yep, that's what I got, too!

tertre  Feb 18, 2018
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7.  The numbers 19, a, b, c, 73 form an arithmetic sequence, in that order.
Find a+b+c.

19 + 4d  =  73       subtract  19  from both sides

4d  =  54   divide both sides by 4

d  =  13.5

Calling m the first term  ....the sum  of a + b + c  is

( m + d ) + (m + 2d) + (m + 3d)  =

3m  + 6d  =

3(19) + 6(13.5)  =

57    + 81  =

138

Feb 19, 2018
#10
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10. The least common multiple of 210 and n is 210n, where n is an integer
greater than 1. What is the smallest possible value of n?

For  210n to be the LCM, n will be the first positive integer that does not share any factors with 210.....this is 11

LCM   210*11  =  2310

Feb 19, 2018
edited by CPhill  Feb 19, 2018
#11
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Wow! Great solution!

tertre  Feb 19, 2018
#12
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11. The side lengths of an acute triangle, in inches, are integers and form
an increasing arithmetic sequence. What is the smallest possible value
for the perimeter of the triangle?

Let m  be the shortest side, m + d the next longest,and m + 2d  the longest

And  we have, by the Triangle Inequality :

( m )  + (m + d) > m + 2d

2m  + d > m + 2d

m > d

Since we have integers, the smallest possible value of d is 1.....which means the smallest  possible value of m must be 2

This means that  the  the second term, m+ d  ≥ 3  and the third term  m + 2d ≥ 4

If the triangle is  2,3,4   we must have the following to be acute

4 < √ [ 2^2 + 3^ 2]

4 < √13       which is untrue

A  3,4,5 triangle is right

If the triangle is  4,5,6  we must have that

6 < √ [ 4^2 + 5^2]

6 < √41  which is true

So....4,5,6   is the triangle we want....and the smallest perimeter is  15

Feb 19, 2018