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27  x  6 ==162 - sum of all 6 numbers

162 - 23 - 16 - 30 - 33 ==60 - sum of the missing 2 numbers.

60 / 2 ==30 

30 + 1==31 - one of the missing 2 numbers

30 - 1 ==29 - the 2nd missing number

So, the 6 numbers are: 23, 16, 30, 33, 29, 31 ==162 - their total

162 - 16 ==146 / 5 ==29.20 - the average of the remaining 5 after removing the smallest.

If you remove all the multiples of 2, 3, 5 and 7 and double counting, you should get:

461 integers, between 1 and 2020,  that are NOT divisible by 2, or by 3, or by 5, or by 7.

3x+7=6x-2

Subtract 3x from both sides

7=3x-2

Add 2 to both sides

3x=9

Divide both sides by 3

x=3

18 ==F  +  4D............................(1)

1448 ==32 / 2 * [2F + 31D......(2), solve for D, F

F==196 / 23 - this is the first term

D ==109 /46 - this is the common difference.

Your sequence looks like this:

8.52173913 , 10.89130435 , 13.26086957 , 15.63043478 , 18 , 20.36956522 , 22.73913043 , 25.10869565 , 27.47826087 , 29.84782609 , 32.2173913 , 34.58695652 , 36.95652174 , 39.32608696 , 41.69565217 , 44.06521739 , 46.43478261 , 48.80434783 , 51.17391304 , 53.54347826 , 55.91304348 , 58.2826087 , 60.65217391 , 63.02173913 , 65.39130435 , 67.76086957 , 70.13043478 , 72.5 , 74.86956522 , 77.23913043 , 79.60869565 , 81.97826087 , >>Number of terms = 32>>Total Sum == 1448

The 9th term ==632 / 23 ==27.47826087

$12,450 cannot be right because it gives a negative value for the horse. Exactly the same question on this forum gives $10,450 plus the horse. I shall use this latter amount:

Let the net worth of the horse==H

[19,200 + H] / 12 ==[10,450 + H] / 7, solve for H

Cross multiply

134,400 + 7H ==125,400 + 12H

134,400 - 125,400 ==12H - 7H

9,000 ==5H

H ==9,000 / 5 ==$1,800 - Net worth of the horse.

1 = 12
2 = 15
3 = 18
4 = 21
5 = 51
6 = 81
7 = 102
8 = 105
9 = 108
10 = 111
11 = 114
12 = 117
13 = 120
14 = 123
15 = 126
16 = 129
17 = 132
18 = 135
19 = 138
20 = 141
21 = 144
22 = 147
23 = 150
24 = 153
25 = 156
26 = 159
27 = 162
28 = 165
29 = 168
30 = 171
31 = 174
32 = 177
33 = 180
34 = 183
35 = 186
36 = 189
37 = 192
38 = 195
39 = 198
40 = 201
41 = 210
42 = 213
43 = 216
44 = 219
45 = 231
46 = 261
47 = 291
48 = 312
49 = 315
50 = 318
51 = 321
52 = 351
53 = 381
54 = 411
55 = 414
56 = 417
57 = 441
58 = 471
59 = 501
60 = 510
61 = 513
62 = 516
63 = 519
64 = 531
65 = 561
66 = 591
67 = 612
68 = 615
69 = 618
70 = 621
71 = 651
72 = 681
73 = 711
74 = 714
75 = 717
76 = 741
77 = 771
78 = 801
79 = 810
80 = 813 - this is the 80th term.

\(A_{45°}=\dfrac{\pi \cdot (2cm)^2\cdot 45°}{360°}=\color{blue}\dfrac{1}{2}\pi\ cm^2\)

  !

For positive integer values of N, let boxed(N) be equal to:
 
boxed(N) = 2+4+6+8+...+N if N is even,
and
boxed(N) = 1+3+5+7+...+N if N is odd.

What is the value of boxed(2009) minus boxed(2008)?

\(\begin{array}{|rcll|} \hline s &=& boxed(2009) \quad minus \quad boxed(2008) \\ s &=& (1 + 3 + 5 + ... + 2009) - (2 + 4 + ... + 2008) \\ s &=& 1 - 2 + 3 - 4 + \dots + 2007 - 2008 + 2009 \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline s &=& 1 - 2 + 3 - 4 + \dots + 2007 - 2008 + 2009 \\ s &=& 2009 - 2008 +2007-2006 + \dots +3-2+1 \\ \hline 2s &=& 2010-2010+2010-2010 + \dots + 2010-2010+2010 \\ 2s &=& 0+0+ \dots +0+2010 \\ 2s &=& 2010 \quad | \quad : 2\\ s &=& 1005 \\ \hline \end{array}\)

boxed(2009) minus boxed(2008)  = \( \mathbf{1005}\)

Let the length of the tunnel ==T 

T + 1 ==80  x  3/60

T + 1 ==4

T ==4 - 1 ==3 miles - length of the tunnel.

Are you SURE that he was paid $12,450? Or was it $10,450? Check your question carefully.

Your highlighted question says this:

" If the population of West Lake is about 11,849 in 2017, the population in 2018 will be about _______. Rounded to the nearest person, the population will be about _____.

Look at the function that is very well explained at the beginning of the question:

P(2018) ==11,849  x  [1 + 2.8/100] ==11,849  x  1.028==12,180.772 ==12,181 - rounded to the nearest person - will be the population of West Lake in 2018.

Note: 2.8% means ==2.8 / 100 ==0.028 

Using the Euclidean thereom, we can find that the largest integer value for n is 234. This question has been posted on several other different platforms as well...

Thanks Heureka,

you are asked for tn  when n = 345

n is a function of k where k is 0 or a positive integer.

in this case k = the floor (whole part) of  345/4   

345/4 = 86 with 1 left over.

k is 86 and 1 is left over

So   

n=4k+1

so tn = 2

think about it and I think you will understand.

The answer is 199.

The only possible values are the first piece is 75, the second 25, and the third 0. That's not 3 pieces, can you piece recheck your question?

Two must be the middle digit here, because if it is the end digit, then it would be even, not odd. So we have 5 numbers to include, 121, 323, 525, 727, and 929. Adding these numbers gets a total value of 2625.

Let s be the length of the smaller square and r be the length of the larger square. Using this, we have the system of equations:

s^2+80=r^2

4s+24=4r

First, we can simplify the second equation to s+6=r, so we know that s=r-6. Plugging this in the first equation gets us

(r-6)^2+80=r^2 --> r^2-12r+116=r^2. Now we now that -12r+116=0. This gives us r=29/3 and s=11/3. Your answer is 44/3 cm.

Please proofread your question.

It is not right.

Do you repost because you are too lazy to finish it yourself? 

I answered this yesterday.

No it is not

Wait... my bad... I made an error. In my original answer, there are 21 terms, not 20. 

It should be \({x+1 \over x} \times {x +2 \over x+1} \times {x +3 \over x +2 } \times \cdot \cdot \cdot \times {x + 20 \over x +19} = 3\). 

This means we have the equation: \(x + 20 = 3x\), where x = 10, meaning the original fraction was \(\color{brown}\boxed{11 \over 10}\)

Yes, I got 23/21 but it was wrong again.