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

area of square ACEG? Express your answer in simplest radical form.

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?

Please answer all questions...Thanks!

mathtoo Feb 18, 2018

#1**+1 **

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**

Guest Feb 18, 2018

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#6**0 **

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

ElectricPavlov Feb 18, 2018

#7**0 **

11. The smallest I can find ...by trial and error is 4 5 6

perimeter = 15

ElectricPavlov Feb 18, 2018

#9**+1 **

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

CPhill Feb 19, 2018

#10**+1 **

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

CPhill Feb 19, 2018

#12**+1 **

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

CPhill Feb 19, 2018