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The given numbers are consecutive prime numbers in increasing order starting with 17.
Hence, the next number in the series is 41.
The given numbers are squares of consecutive natural numbers in decreasing order starting with 15,
i.e., the numbers 225, 196, 169, ____, 100, 81 can be written as 15^{2}, 14^{2}, 13^{2} , ____, 11^{2} , 10^{2} , 9^{2}.
Hence, the missing number is 12^{2} = 144.
1st term = 54
2nd term = 54 + 12 = 66
3rd term = 66 + 16=82
4th term = 82 + 20 =102
5th term = 102 + 24 = 126...
The difference is increasing by 4, starting with 12.
So, the next difference is 24 + 4 = 28.
Hence, the next number is 126 + 28 = 154.
List of Prime Numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,...
The given numbers are alternate prime numbers in decreasing order, starting with 97.
Hence, the next number in the series is 47.
1st term = 8
2nd term = 8 x 2 = 16
3rd term = 16 x 3 = 48
4th term = 48 x 2 = 96
5th term = 96 x 3 = 288
6th term = 288 x 2 = 576...
⇒ For every term at even position, the previous term is multiplied by 2.
⇒ For every term at odd position, the previous term is multiplied by 3.
So the missing term is: 576 × 3 = 1728
1st term = 75
2nd term = 75 + 216 = 75 + 6^{3} = 291
3rd term = 291 + 125 = 291 + 5^{3} = 416
4th term = 416 + 64 = 416 + 4^{3} = 480
5th term = 480 + 27 = 480 + 3^{3} = 507...
The differences are cubes of consecutive natural numbers in decreasing order from 6.
So, the missing term = 507 + 8 = 507 + 2^{3} = 515
The given numbers are consecutive natural numbers in decreasing order starting with 225. Hence, the missing number is 223.
1st term = 5
2nd term = 5 × 2 + 3 = 13
3rd term = 13 × 3 + 2 = 41
4th term = 41 × 2 + 3 = 85
5th term = 85 × 3 + 2 =257...
⇒ For every term at even position, the previous term is multiplied by 2 and 3 is added.
⇒ For every term at odd position, the previous term is multiplied by 3 and 2 is added.
So the missing term is: 257 × 2 + 3 = 517
The common difference between consecutive terms is 6, 12, 6, 12 & so on. So, the term next to 141 is 141 + 6 = 147
The 1st to 5th terms are 0, 5, 22, 57, 116.
These terms are 1 – 1, 8 – 3, 27 – 5, 64 – 7, 125 – 9.
i.e. 1^{3} – 1, 2^{3} – 3, 3^{3} – 5, 4^{3} – 7, 5^{3} – 9
So, the nth term of the series: n^{3} – (2n – 1).
So, the 6th term of the series is 6^{3} – (2 × 6 –1) = 216 – 11 = 205.
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