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## Explanation Number Sequences Hard

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1. It is a generalization of the fibonacci sequence knows as the tetranacci sequence. It is similar to the fibonacci sequence where each subsequent number is the sum of the previous two. With a tetranacci sequence each subsequent number is the sum of the previous four, so the full sequence looks like this: 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401. This sequence can be very hard if you have no idea were to look, therefore learn to recognise these types of sequences. However…

There are often more answers correct to most likely answer to this sequence as it is presented is probably 400.

29 = 2 x 15 -1
56 = 2 x 29 – 2
108 = 2 x 56 – 4
208 = 2 x 108 – 8
400 = 2 x 208 -16

2. This sequence consists out of two alternating series.

13   -21   34   -55   89   -144

1. The first series shows the relation between the 1st and 3rd number, 3rd and 5th etc. in this case this following relation exists: 34-13=21. The next number in the sequence is -21.
2. The second series shows the relation between the 2nd and 4th number, the 4th and 6th etc. in this case the difference between -55 and -21 =34. The next number in the sequence is -34.
3. The next number in the sequence will thus be -55-89=-144.

3. This sequence consists out of two alternating series.

52   56   48   64   32   96

1. The first series shows the relation between the 1st and 3rd number, 3rd and 5th etc. in this case this following relation exists: 52-48=4 48-32=16. The difference is a factor of 4.
2. The second series shows the relation between the 2nd and 4th number, the 4th and 6th etc. in this case 64-56=8 and the unknown number which has to be filled in.
3. From the first series it is known that this difference is a multiple of 4, thus in this case that results in 8*4=32. 64+32=96.

4. The pattern in this sequence is +2,+4, +4 after which this pattern repeats itself.

1   3   7   11   13   17

+2   +4   +4   +2   +4

5. This sequence consists out of two alternating series.

230   460   46   92   9.2   18.4

1. The first series shows the relation between the 1st and 3rd number, 3rd and 5th etc. in this case this following relation exists: 230/46=5 46=9.2=5
2. The second series shows the relation between the 2nd and 4th number, the 4th and 6th etc. in this case 460/92=5
3. This means the series are decreasing with a factor of 5 with increasing sequence. The next number in the sequence will thus be 92/5=18.4

6. Alternating addition with increasing sequence.

3   12   24   33   66   75

+9   +12   +9   +33   +9

7. This is really hard to spot if you don’t see it at once and also requires some of the box thinking. This sequence consists of the first prime numbers. Therefore the correct answer is the next prime number in the sequence is this is “17”.

3   5   7   11   13   17

8. This sequence consists of square numbers, but very other number is subtracted by 1. So the first one is the square of 3 minus 1, the second number is just the square of 4 etc. The correct answer would be the square of 8 (8*8) = 64.

8   16   24   36   48   64

32-1    42    52-1    62    72-1

9. The pattern in this sequence is as follows: First the number is divided by 2 after which 2 is added to the result.

68   36   20   12   8   6

:2+2   :2+2   :2+2   :2+2   :2+2

10. Alternating dividing or multiplying with increasing sequence.

18   6   24   8   32   10.67

:3   *4   :3   *4   :3

11. This sequence is constructed in the following way:

99   18   36   9   18   9

1. 99 can be seen as 9+9=18 (which is the 2nd presented number)
2. 36 can be seen as 3+6=9 (the 4the number)
3. 18 can thus be seen as 1+8=9 which is the next number in row.

12. This sequence consists out of two alternating series.

3   8   23   68   203   608

1. The first series shows the relation between the 1st and 3rd number, 3rd and 5th etc. in this case this following relation exists: 23-3=20 203-23=180.
This difference is a multiple of each other by a factor of 180/20=9.
2. The second series shows the relation between the 2nd and 4th number, the 4th and 6th etc. in this case 68-8=60. Finding the next number in the sequence means using the relation from the first series.
3. This means that 60 must be multiplied by 9, resulting in 540. This number is added to 68 which results in 608 as next number in the sequence.

13 The pattern in this sequence is as follows: First the number is divided by 2 after which 1 is added to the result.

144   73   14   8   236   119

:2+1   :2+1   :2+1   :2+1

14 This sequence consists out of two alternating series.

10   45   15   38   20   31

1. The first series shows the relation between the 1st and 3rd number, 3rd and 5th etc. in this case this following relation exists: 10-15-20 with an addition of 5 every time.
2. The second series shows the relation between the 2nd and 4th number, the 4th and 6th etc. in this case 45-38=7. Here the relation is a subtraction of 7.
3. The next number in the sequence will thus be 38-7=31

15 The difference increases with a multiple of 9 with increasing sequence.

1   10   37   82   145   226

+9   +3*9   +5*9   +7*9   +9*9

16 The difference increases with +2 with increasing sequence.

-2   1   6   13   22   33

+3   +5   +7   +9   +11

17 The sequence is being constructed in the following way:

34   -21   13   -8   5   -3

1. 34-13=21. This is the second number in the sequence in a negative form.
2. -21+13=-8. This is the third number in the sequence.
3. Continuing results in -8+5=-3.

18. The following sequence is an example of a negative fibonacci sequence.

1   0   1   -1   2   -3

1-0=1,   0-1=-1,   1-(-1)=2,   -1-2=-3

19 This question has a link with the 1st question of this hard test and if you noticed the similarities, than you could have used this information to solve the sequence:

108   56   29   15   8   4

If you look closer then you can see it is actually the 1st question in reverse. If you would combine the numbers then it might give you an edge to solve it. You could see this as a reverse tetranacci sequence. Yes I know… However this is the hard test

0,  0,  0,  1,  1,  2,  4,  8,  15,  29,  56,  108,  208,  401

20 The pattern in this sequence is as follows:

1   8   27   64   125   216

(=13)   (=23)   (=33)   (=43)   (=53)   (=63)

1. Raise the number to the power of 3
2. Subtract 1 from the result.
3. The numbers which need to be raised to the power increase with 1 with increasing sequence.

21.

-2   3   27   69   129   207

=1*3   =9*3   =23*3   +43*3   =69*3

This sequence is constructed out of multiples of 3 in the following way:

1   9   23   43

+8   +14   +20   +26

The difference in the multiples increases with 6 with increasing sequence. This means that the next difference 26 is. The next step is the addition of 26 to 43, resulting in 69 which is multiplied by 3 to result in the answer of the sequence 207.