Number Series

The series follows an alternating pattern of adding 1, subtracting 4, adding 9, subtracting 14, adding 23, subtracting 36. Alternatively, looking at differences: +1, -4, +9, -14, +23, -36. The sequence of differences is 1^2, -2^2, 3^2, -4^2, 5^2, -6^2. Applying this, 50+1=51, 51-4=47, 47+9=56, 56-16=40 (not 42), 40+25=65, 65-36=29. Thus, 42 is the incorrect term.

The series is 64, 65, 61, 70, 54, ... The differences are +1, -4, +9, -16. The next difference should be +25. 54 + 25 = 79.

145 - 110 = 35. 110 - 82 = 28. 82 - 61 = 21. 61 - 47 = 14. The differences are 35, 28, 21, 14 (multiples of 7). The next difference should be 7. 47 - 7 = 40.

The pattern is (previous number * 3) + 2. 9*3+2=29, 29*3+2=89, 89*3+2=269, 269*3+2=809, 809*3+2=2429.

The series is 13 (+12) = 25, 25 (+24) = 49, 49 (+36) = 85, 85 (+48) = 133, 133 (+60) = 193. The term 84 is incorrect; it should be 85.

Pattern: x*2 + 1, x*2 + 2, x*2 + 3, x*2 + 4... 3*2+1=7, 7*2+2=16, 16*2+3=35, 35*2+4=74, 74*2+5=153. The term 152 is incorrect; it should be 153.

The pattern is: 11*2-7=15, 15*2-7=23, 23*2-7=39, 39*2-7=71, 71*2-7=135 (not 139), 135*2-7=263. The wrong number is 139.

The pattern is +5, -3, +5, -3, +5. 12 + 5 = 17.

The pattern is subtracting consecutive integers starting from 2: 72-2=70, 70-3=67, 67-4=63 (not 65), 63-5=58, 58-6=52, 52-7=45. The number 65 is incorrect as it should be 63.

The pattern is: 15*1+1=16, 16*2+2=34, 34*3+3=105, 105*4+4=424, 424*5+5=2125. The term 2124 is incorrect; it should be 2125.

The series is 17, 19, 15, 23, 7, 39. The pattern alternates: +2, -4, +8, -16, +32. The next step is -64. 39 - 64 = -25.

The series is 2, 6, 12, 20, 30... which is n(n+1). Sum = sum(n^2 + n) = sum(n^2) + sum(n) = [n(n+1)(2n+1)/6] + [n(n+1)/2]. For n=30: [30*31*61/6] + [30*31/2] = 9455 + 465 = 9920.

The differences between consecutive terms are: 529 - 521 = 8 (2^3), 556 - 529 = 27 (3^3). The next difference should be 4^3 = 64. 556 + 64 = 620. Checking the next: 745 - 620 = 125 (5^3), which confirms the pattern.