Number Series

The pattern is: 5*2-1=9, 9*2-2=16, 16*2-3=29, 29*2-4=54, 54*2-5=103, 103*2-6=200.

The series follows two alternating patterns: (2, 3, 4, 5) and (0, 1, 2, 3). The missing term is the next in the first sequence, which is 5.

Pattern: 2*2+1=5, 5*3+1=16, 16*2+1=33, 33*3+1=100, 100*2+1=201, 201*3+1=604. The sequence is alternating *2+1 and *3+1. 98 is incorrect; it should be 100.

The differences are: 325-259=66, 259-204=55, 204-160=44, 160-127=33, 127-105=22. The next difference should be 11. 105-11 = 94.

Differences: 100-95=5, 117-100=17, 154-117=37. Second differences: 12, 20. Third difference is 8. Next second difference = 20+8=28. Next first difference = 37+28=65. 154+65=219.

The series follows the pattern: 812 / 2 - 8 = 398; 398 / 2 - 7 = 192; 192 / 2 - 6 = 90; 90 / 2 - 5 = 40; 40 / 2 - 4 = 16. The missing term is 192.

The recurrence S_n = S_{n-1} - 10 is an arithmetic progression with common difference -10. S_3 = 0, so S_n = 0 - 10(n - 3) = 30 - 10n. Terms: S_10 = -70, S_20 = -170, S_30 = -270, S_40 = -370. Sum = -70 - 170 - 270 - 370 = -880.

The series is (1^2 - 2^2) + (3^2 - 4^2) + ... + (99^2 - 100^2). Each pair (n^2 - (n+1)^2) equals -(2n+1). There are 50 such pairs, so the sum is -(3 + 7 + 11 + ... + 199). This is an arithmetic progression with 50 terms, sum = (50/2) * (3 + 199) = 25 * 202 = 5050. Thus, the total is -5050.

The differences between consecutive terms are 8, 12, 16, 20. These differences increase by 4 each time, so the next difference should be 24. Adding 24 to 59 gives 83.

The sequence follows the pattern n^2 + 1: 1^2+1=2, 2^2+1=5, 3^2+1=10, 4^2+1=17, 5^2+1=26, 6^2+1=37. The next term is 7^2 + 1 = 50.

Sequence 1: 17, 21, 25, ..., 817 (a=17, d=4). Sequence 2: 16, 21, 26, ..., 851 (a=16, d=5). Common terms start at 21 and increase by LCM(4,5) = 20. Common sequence: 21, 41, ..., up to 817. 817 = 21 + (n-1)20. 796 = (n-1)20. 39.8 = n-1, so n = 40.

The series follows the rule: each number is the sum of the two preceding numbers (Fibonacci-like). 4+7=11, 7+11=18, 11+18=29, 18+29=47, 29+47=76.

The pattern is (previous term / 2) + 4. 1248/2 + 4 = 628. 628/2 + 4 = 318. 318/2 + 4 = 163 (not 165). 163/2 + 4 = 85.5. 85.5/2 + 4 = 46.75. 46.75/2 + 4 = 27.375. The wrong number is 165.