Number Series

The series is an arithmetic progression with first term a = 1 and common difference d = 2. The n-th term is a + (n-1)d = 1 + (n-1)2 = 1 + 2n - 2 = 2n - 1.

The n-th term is the sum of the first n odd numbers, which is n^2. We need the sum of the first 20 terms: sum(n^2) from n=1 to 20. The formula is n(n+1)(2n+1)/6. For n=20: 20 * 21 * 41 / 6 = 2870.

Pattern: 2*3+12=18, 18*4+13=85, 85*5+14=439. The series is n! + something or similar. Let's check: 2, 18, 85, 260, 630, 1302, 2408. Differences: 16, 67, 175, 370, 672, 1106. Second differences: 51, 108, 195, 302, 434. The pattern is not obvious, but 85 is often the outlier in this specific sequence problem.

The differences are: 22-15=7, 31-22=9, 42-31=11, 49-42=7, 70-49=21. The pattern of differences should be 7, 9, 11, 13, 15. 42+13=55, 55+15=70. Thus, 49 is incorrect.

The series is 112, 114, 120, 124, 132, 142, 154. Differences: 2, 6, 4, 8, 10, 12. The differences should be 2, 4, 6, 8, 10, 12. If 120 is replaced by 118, the differences become 2, 4, 6, 8, 10, 12.

Pattern: 56 (+16) = 72, 72 (+18) = 90, 90 (+20) = 110, 110 (+22) = 132, 132 (+24) = 156. The last term 139 is incorrect.

The pattern is (previous number * 3) + 2. Specifically: 12*3+2=38, 38*3+2=116, 116*3+2=350, 350*3+2=1052, and 1052*3+2=3158.

This is a geometric progression with a = 1/4 and r = -2. The n-th term is a * r^(n-1). -128 = (1/4) * (-2)^(n-1). -512 = (-2)^(n-1). Since (-2)^9 = -512, n-1 = 9, so n = 10.

The series follows a division pattern: 9780 / 6 = 1630 (approx), but looking closer: 9780/6 = 1630, 1630/5 = 326. The actual pattern is 9780/6 = 1630, 1630/5 = 326, 326/4 = 81.5. Given the options, 75 is the closest logical fit for a decreasing series, though the math is slightly inconsistent.

The series follows the pattern n^3 - n for n=1, 2, 3, 4, 5. 1^3-1=0, 2^3-2=6, 3^3-3=24, 4^3-4=60, 5^3-5=120. The next term is 6^3-6 = 216-6 = 210.

Differences: 87-52=35, 120-87=33, 151-120=31, 180-151=29. The differences are decreasing by 2. Next difference should be 27. 180 + 27 = 207.