17, 19, 23, 29, 31, 37, _____
The given numbers are consecutive prime numbers in increasing order starting with 17.
Hence, the next number in the series is 41.
225, 196, 169, _____, 121, 100, 81
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.
54, 66, 82, 102, 126, _____
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.
97, 83, 73, 67, 59, _____
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.
8, 16, 48, 96, 288, 576, _____
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
75, 291, 416, 480, 507, ____
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
225, 224, _____, 222, 221
The given numbers are consecutive natural numbers in decreasing order starting with 225. Hence, the missing number is 223.
5, 13, 41, 85, 257, _____
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
123 , 129, 141, _________, 159, 165
The common difference between consecutive terms is 6, 12, 6, 12 & so on. So, the term next to 141 is 141 + 6 = 147
0, 5, 22, 57, 116, _________
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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