Number Series

Differences: 1, 4, 9, 16. These are squares: 1^2, 2^2, 3^2, 4^2. The next difference should be 5^2 = 25. 35 + 25 = 60.

Pattern: 105 - 20 = 85; 85 - 25 = 60; 60 - 30 = 30; 30 - 35 = -5 (not 0); -5 - 40 = -45; -45 - 45 = -90. The term 0 is wrong; it should be -5.

This is an arithmetic progression with first term a = 357 and common difference d = 6. The 10th term is calculated as a + (10 - 1)d = 357 + 9(6) = 357 + 54 = 411.

This is an arithmetic progression with a=5, d=3. Formula: a_n = a + (n-1)d. 320 = 5 + (n-1)3. 315 = (n-1)3. 105 = n-1. n = 106.

The pattern is: 5 + 4 = 9; 9 + 7 = 16; 16 + 13 = 29; 29 + 25 = 54. The differences are 4, 7, 13, 25. These follow the pattern (diff * 2 - 1): 4*2-1=7, 7*2-1=13, 13*2-1=25, 25*2-1=49. 54 + 49 = 103.

The series is n^3 + 1: 0^3+1=1, 1^3+1=2, 2^3+1=9, 3^3+1=28, 4^3+1=65, 5^3+1=126. The missing term is 3^3+1 = 28.

The series follows the pattern n^3 - 1. 2^3-1=7, 3^3-1=26, 4^3-1=63, 5^3-1=124, 6^3-1=215, 7^3-1=342. The next term is 8^3-1 = 512-1 = 511.

The differences are: 9-7=2, 17-9=8, 42-17=25, 91-42=49, 172-91=81, 293-172=121. The pattern of squares is 1^2, 3^2, 5^2, 7^2, 9^2, 11^2. The difference 2 is wrong; it should be 1^2=1 (making the second term 8) or the first term should be different.

This is a geometric series with a=1, r=3. Sum = a(r^n - 1) / (r - 1). 1093 = 1(3^n - 1) / 2. 2186 = 3^n - 1. 2187 = 3^n. Since 3^7 = 2187, n = 7.

The series is an arithmetic progression with a common difference of 10. The next term is 25 + 10 = 35.

The series follows the pattern n^2 - 1 for n=2, 3, 4, 5, 6, 7, 8. Specifically: 2^2-1=3, 3^2-1=8, 4^2-1=15, 5^2-1=24, 6^2-1=35, 7^2-1=48, 8^2-1=63. The term 34 is incorrect as it should be 35.

The series follows a pattern of alternating operations: divide by 2, then add 2. Starting from 100: 100/2 = 50, 50+2 = 52, 52/2 = 26, 26+2 = 28, and finally 28/2 = 14.

The pattern is division: 19200/10 = 1920; 1920/8 = 240; 240/6 = 40; 40/4 = 10; 10/2 = 5.