Given series 0, 2, 8, 14, ___,34?

1² -1=0

2² -2=2

3² -1=8

4² -2=14

5² -1=24

6² -2=34

Answer is 24,

First term : 4
Second term: 4+3 = 7
Third term : 7+5 = 12
Fourth term : 12+7 = 19.
Fifth term : 19+9 = 28.
Therefore,
Next Terms : 28+11 = 39

The sequence is a combination of following two series:
I. 3, 8, 13, 18, 23 and II. 12, 19, 32, 42, 52
The pattern in I has a constant difference of +5, and the pattern in II has a constant difference of +10. So, in II, 19 is wrong and should be replaced by (12+10) i.e. 22

The pattern is :

1²=1

3²=9

5²=25

7²=49

9²=81

11²=121

as we can see that when 6*4=24 then 5*5=25 then again 24*6=144,hence 25*7=175 . there is a constant multiplication of x+1 where initially x is 4 then 5 then 6 and at last 7

The difference between two consecutive numbers is increasing in the order of 2,3, 4, 5, etc.

This is a simple alternating addition and subtraction series.
In the first pattern, 3 is added;
In the second pattern, 2 is subtracted.

Rule =(1st number) (1st number + 3 = 2nd number) (2nd number – 2 = 3rd number) (3rd number + 3 = 4th number)
Now (7) (7 + 3 = 10) (10 – 2 = 8) (8 + 3 = 11) (11 – 2 = 9) (9 + 3 = 12) (12 – 2 = 10)…
So the answer is b = 10.

18 + 10 = 28

28 + 12 = 40

40 + 14 = 54

54 + 16 = 70

70 + 18 = 88

? = 88

This is an alternating repetition series. The number 32 alternates with a series in which each number decreases by 2.

Let us first closely examine the sequence for any mathematical correlation between the numbers of sequence

It is visualized that sequence, is progressing with multiplication of ×2 in the number of the sequence then addition of increasing consecutive number starting from +1 (viz 1, 2, 3, 4, 5, 6…) in the previous number of the sequence to obtain next number of the sequence.

The given sequence is

1, 3, 8, 19, 42, 89, ___?

Let us see from first number onward

1 × 2 + 1 = 3

3 × 2 + 2 = 8

8 × 2 + 3 = 19

19 × 2 + 4 = 42

42 × 2 + 5 = 89

89 × 2 + 6 = 184

184 × 2 + 7 = 375

375 × 2 + 8 = 758

The extended sequence is

1, 3, 8, 19, 42, 89, 184, 375, 758

Answer the next numbers in the sequence are 184, 375 and 758

Explanation:
There are two series (8, 11, 14, 17, 20) and (7, 12, 17, 22) increasing by 3 and 5 respectively.

Explanation:

8 – 7 = 1
16 – 8 = 8
43 -16 = 27
1, 8, 27 are cubes of 1, 2 and 3
So next difference must be cube of 4
So, ? – 43 = 43 = 64
So, ? = 64+43 = 107

Explanation:
The given series follows a logic that

11 x 12, 12 x 13, 13 x 14, 14 x 15, 15 x 16,…

So the missing number is 13 x 14 = 182

184

276

12 = 1.
32 = 9.
52 = 25.
72 = 49.
92 = 81.
112 = 121.

This is an alternating addition and subtraction series. Roman numbers alternate with Arabic numbers. In the Roman numeral pattern, each number decreases by 1. In the Arabic numeral pattern, each number increases by 1.

This is a simple subtraction series. Each number is 35 less than the previous number.

This is a simple alternating addition and subtraction series. In the first pattern, 3 is added; in the second, 2 is subtracted.

Hence, 7 +3 10 -2 8 +3 11 -2 9 +3 12 -2 10

In this alternating repetition series, a random number, 33, is interpolated every third number into a simple addition series, in which each number increases by 2.

Here we see ups and downs in the series so there is a chance of 2 separate series
We can have 20 24 30 38 as one series
And 4 64 196 as other series.
Both series occur alternately so next term has to be of the second series.
In first series difference is
24 – 20 = 4
30 – 24 = 6
38 – 30 = 8
So, difference increases by 2
In second series, We see they are squares i.e. 2², 8², 14²
8-2 = 6
14-8 = 6
So next number, ? – 14 has to be = 6
So, ? = 14+6 = 20
So next term is 20² = 400

First term → 9

Second term → (9 *2 -1) = 17

Third term → (17 *2 -1) = 33

Fourth term → (33 *2 -1) = 65

Fifth term → (65 *2 -1) = 129

496 – 492 = 4 = 2²
492 – 483 = 9 = 3²
483 – 467 = 16 = 4²
So, the next has to be a difference of 25 = 5²
467 – 25 = 442

In this simple subtraction series, the numbers decrease by 3.

5
5 x 3 + 6 = 21
21 x 3 + 6 = 69
69 x 3 + 6 = 213
213 x 3 + 6 = 645
645 x 3 + 6 = 1941

This is a simple addition series with a random number, 8, interpolated as every other number. In the series, 6 is added to each number except 8, to arrive at the next number.