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Nevermind I got it!!!

5

Σ    22   +  5n

n = 0

34  can be written as    3b + 4

15 can be written  as    1b +  5

554 can be written as  5b^2 + 5b + 4

So we have  that

(3b  + 4)  ( 1b + 5)  =  5b^2 + 5b  +  4

3b^2  + 19b + 20  = 5b^2 +  5b + 4     simpify as

2b^2 - 14b - 16  =  0      

b^2  - 7b - 8  =  0    factor

(b - 8) ( b + 1)   = 0

Only  b  - 8   provides a solution 

b - 8   =  0

b  = 8

When x = 0 , y = 18

When  x = 9, y  = 0

The line passes through a lattice point  for every two  units it rises from right to left

So......the number of lattice points   is   1  + [ 18 - 0 ]  / 2   =   1 + 18/2  =   1 + 9      =  10

OK, MoonWolfie....I forgot to sum the  coordinates, but I suppose you  can handle that....

Using the triangle inequality

33 + 42  >   than the remaining side  length

75 >  than the remaining side length

So.......the max length for the remaining side  =  74

And the max perimeter =   33 + 42 + 74  =    149 

Thank you so much!!

Not as difficult as it seems once we  draw  a graph :  https://www.desmos.com/calculator/wars2lfhv7

Construct a circle with a center of (7,13)  and  a  radius of 13

And consider the line y  = 13

This line will pass through (7,13)

The equation of our circle is

(x-7)^2  + ( y - 13)^2  =  169

Let y = 18   ( this will be 5 vertical  units  from y = 13 )

(x - 7)^2  + (18 - 13)^2  =  169

(x - 7)^2  + 5^2 =  169

(x - 7)^2  + 25 =  169

(x - 7)^2  =  169 -25

(x -7)^2  = 144

12^2 = 144          and          (-12)^2  =  144

So  x must be  19       or      x =  -5

And one of  the points we  need is  ( 19, 18)This point will lie on our circle and will be 13 units from (7 ,13)  and it will be 5 units  from the line  y = 13

And one ot the other points we  need is  (-5, 18)

And using symmetry.....the other two points we  need are     (19, 13 - 5)  = (19, 8)     and (-5,13-5) = (-5,8)

2. A restaurant mixes 2 gallons of milk containing 1% fat and 3 gallons of milk containing 2% fat. What is the percent of fat in the mixture? Express your answer to the nearest tenth of a percent.

1%  of 2  =  .02  gallons of fat

2% of 3 =    ,06 gallons of fat

% fat in the mixture  =    [ .02 + .06 \  /  [ 2 + 3 ] =      .08 / 5  = .016  =  1.6% 

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1.  Look at the eding digits  for powers of 7

7^1  =  7

7^2 =   9

7^3  =  3

7^4  =  1

And this pattern repeats  for every  4 powers of 7

So   2008 mod 4  =  0

So....the units digit for  2007 ^2008   =  1

Likewise.....look at the ending digits for  powers  of 8

8^1  =  8

8^2 =  4

8^3  =  2

8^4  =  6

This also repeats for  every  4 powers of 8

So....2007  mod 4  = 3

The units digit for 2008^2007  =  2

So.....the  ending units digit  for 2007^2008 + 2007^2008 =  1 + 2   =  3

1 - First term = 3 / 49

2 - The value of r = 13 / 49

3 - The value of r tells you that the series converges. In fact, it converges to = 1 / 12

4 - Here are the first 5 - 6 terms of the series: 3/49,   39/2401,   507 / 117649,   6591 / 5764801,   85683 / 282475249,   1113879 / 13841287201...etc.

(2x^5 - (1/9)y^4)^5

a)  Summation notation

5

Σ  C(5,n)   (2x⁵)^(5-n) (-(1/9)y⁴)ⁿ

n = 0

(b)   3rd term =

C (5,2) * (2)³ x¹⁵ * (-(1/9y⁴)²  =    10  * (8x¹⁵) (1/81)y⁸    =  (80/81)x¹⁵ y ⁸

NOTE: that other answer was a bit INCORRECT    the common ratio is MINUS 6/10      NOT + 6/10

That makes sense, thank you!

See answer here:  https://web2.0calc.com/questions/given-these-terms-in-a-geometric-series-160-96-288-5-864-25#r1

Sorry, your question does not translate very well on "Google Translate". But let me see if I understand your question!.

Beatrice, Laurence and Marianne together have: $2,561.

Marianne has x dollars.

Beatrice has 4x dollars.

Laurence has 2 x 4x =8x dollars. So:

x + 4x + 8x =2,561

13x =2,561

x =2,561 / 13 = 197 dollars - this is what Marianne has.

197 x 4 =788 dollars - this is what Beatrice has.

197 x 8 =1,576 dollars - this is what Laurence has.

Check: 197 + 788 + 1,576 =2,561 dollars. I hope I understood your question.

3x+5 is always going to be odd given that x is even positive integer (check it and see if you can figure out why), therefore the largest odd number less than it, is (3x+5)-2 = 3x+3

In a similar way, 7x-2 is always going to be even, therefore the number immediately next to it the right is the smallest odd number we can find bigger than it, which we can express as (7x-2)+1 = 7x-1

Now we can just find the difference between the two expressions we figued out. 

7x -1 - (3x+3) = 7x -1 -3x -3 = 4x -4

"Distinct 4-digit numbers". I think you only have: 1129, 1136, 1233. The other 33 listed above are simply "permutations" on these 3 "distinct combinations".

105  175  245  315  385  455  525  595  665  735  805  875  945  Total =  13 such numbers.

Maybe you can see here: https://web2.0calc.com/questions/i-got-8-possibilities-it-is-incorrect and derive the answer from that...

1129  1136  1163  1192  1219  1233  1291  1316  1323  1332  1361  1613  1631  1912  1921  2119  2133  2191  2313  2331  2911  3116  3123  3132  3161  3213  3231  3312  3321  3611  6113  6131  6311  9112  9121  9211  Total =  36 such numbers.

To be parallel and tangent, the lines must be on opposite sides of the diameter.

    they will be perpindicular to that diameter line which has slope  (5-12) /(2-9) =1

       the perpindicular slope to this diameter is the slope of the tw lines = - 1/1 = -1

Here is a picture:

Thanks Alan. Great idea! So:

13^4 - 12^4 =7,825 - 4-letter "words" that have at least one "A"

13^3 - 12^3 = 469  -  3-letter "words"  that have at least one "A"

13^2 - 12^2 = 25    -   2-letter "words" that have at least one "A"

13^1 - 12^1 = 1       -    1-letter "word" that has one "A"

7,825 + 469 + 25 + 1 =8,320 "words" that have at least one "A"

My reasoning is as follows:

There are 13^4 possible four letter words in total that can be made with 13 letters.

There are 12^4 possible four letter words that can be made with the 12 letters that aren't A.

Hence there are 13^4 - 12^4 four letter words that contain at least one A.  (13^4 - 12^4 = 7825).

We then need to use similar reasoning for one two and three letter words.