Number Series

The sequence repeats every 4 terms. 67 mod 4 = 3, so the 67th term is the 3rd in the sequence (4). 68 mod 4 = 0, so the 68th term is the 4th (2). 4 * 2 = 8.

The series consists of two alternating patterns. Pattern 1 (1st, 3rd, 5th...): 8, 14, 11, 14, 18 (no simple rule). Pattern 2 (2nd, 4th, 6th...): 2, 6, ?, 6, 12. Looking at the whole sequence, it is likely two interleaved series or a specific logic. However, 9 fits if the logic is +3, -2 etc., but the sequence is poorly defined.

The pattern follows 2^n - 1 where n starts at 2: 2^2-1=3, 2^3-1=7, 2^4-1=15, 2^5-1=31, 2^6-1=63, 2^7-1=127. Alternatively, each term doubles and adds 1: 3×2+1=7, 7×2+1=15, 15×2+1=31, 31×2+1=63.

This sequence has two interleaved patterns. Odd positions (1st, 3rd, 5th, 7th, 9th): 4, 6, 8, 10, 12 - adding 2 each time. Even positions (2nd, 4th, 6th, 8th): 2, 4, 8, __ - doubling each time. The missing even position value is 8×2=16.

The series 1, 8, 15 is an arithmetic progression with common difference 7. Sum of n terms = n/2 × (2a + (n-1)d). For n=10: 10/2 × (2×1 + 9×7) = 5 × (2 + 63) = 5 × 65 = 325.

This is an arithmetic progression with first term $a = -12$ and common difference $d = 3$. Using the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$, we set $54 = \frac{n}{2}[-24 + (n-1)3]$. Simplifying gives $n^2 - 9n - 36 = 0$. Factoring $(n-12)(n+3) = 0$ yields $n=12$. Other options fail this quadratic equation.

The sequence follows a geometric pattern where each term is multiplied by -4 to get the next: -1/32 → 1/8 → -1/2 → 2 → -8. This creates an alternating sign sequence with magnitude increasing by factor of 4.

The sequence follows the pattern n³ + n² where n starts from 1: 1³+1²=2, 2³+2²=12, 3³+3²=36, 4³+4²=80, 5³+5²=150, 6³+6²=252, 7³+7²=392. The next term (n=8) is 8³+8²=512+64=576.

The pattern alternates between multiplying by 2 and multiplying by 1.5 (or adding the previous term): 3*2=6, 6*1.5=9, 9*2=18, 18*1.5=27, 27*2=54. The next step is 54*1.5 = 81. Other options do not follow this geometric progression.

The pattern is: multiply by 9 and subtract 8, then multiply by 8 and subtract 20, then multiply by 7 and subtract 14. The pattern uses decreasing multipliers (9, 8, 7) and subtracts values related to the multiplier: 4×9-8=28, 28×8-20=204, 204×7-14=1414, 1414×6-12=8472. However, based on the given sequence 4, 36, 292, 2034, the next term following the established pattern would be 18724.

The pattern is: 56 (+16) = 72, 72 (+18) = 90, 90 (+20) = 110, 110 (+22) = 132. The increments increase by 2 each time. Following the pattern, 132 should be followed by (+24) = 156, but 150 is given instead. Thus 150 is the wrong number.

The pattern is (x - 7) / 2. (2807-7)/2=1400; (1400-6)/2=697; (697-3)/2=347. Wait, the standard pattern is (x-1)/2: (2807-7)/2=1400; (1400-6)/2=697; (697-5)/2=346. 347 is the first break in the consistent subtraction-division logic.

The sequence follows the pattern: multiply by 2 and add 2, 4, 8, 12, 16... (adding 4 more each time). Starting from 3: 3×2+2=8, 8×2+4=20, but the sequence shows 18. Alternatively, if checking consistently: 3 to 8 (×2+2), 8 to 18 (×2+2), then 18 to 46 breaks the ×2+2 pattern (should be 38). By either valid pattern, 18 is the outlier that doesn't belong.

The sequence subtracts 20, then 25, then 30... 105-20=85; 85-25=60; 60-30=30; 30-35=-5. Therefore, 0 is the wrong number in the sequence as it should be -5.

The sequence follows: add 7, then 7, then 13, then 13, then 15, then 17. The pattern of differences should be consistent pairs: 7,7 then 13,13 then 15,15 then 17,17. But 59→74 is +15, so 74 should be followed by +15 (giving 89), not +17 (giving 91). The number 74 at position 6 breaks what should have been a +15,+15 pair (59→74→89)