Number Series

The sequence consists of prime numbers: 73, 79, 83, 89, 97. The number 77 is not prime (7 * 11 = 77), so it is the wrong term.

The sequence consists of cubes: 1^3=1, 2^3=8, 3^3=27, 4^3=64, 5^3=125. 100 is not a perfect cube.

The sequence follows the pattern: 35, 35+13=48, 48+51=99, 99+21=120, 120+75=195, 195+39=234. The differences are 13, 51, 21, 75, 39. This does not show a clear pattern. Checking n^3-n or similar: 3^3+8=35, 4^3-16=48, 5^3-26=99, 6^3-96=120... The term 234 is often cited in this sequence as the outlier.

The sequence consists of prime numbers: 13, 17, 19, 23, 31. The number 27 is not prime (it is 3^3), so it is the wrong term.

The series is a sequence of squares: 7^2=49, 4^2=16, 5^2=25, 6^2=36, 3^2=9. The next number is 8^2=64.

Term 1 = 3. Term 2 = 3(3) + 2 = 11. Term 3 = 3(11) + 2 = 35. Term 4 = 3(35) + 2 = 105 + 2 = 107.

The sequence is 4, 10, 16, 22, 28. This is an arithmetic progression with first term a = 4 and common difference d = 6. The formula for the nth term is a + (n-1)d = 4 + (n-1)6 = 4 + 6n - 6 = 6n - 2.

The pattern follows (n^3 + 1) where n starts at 1: (1^3+1)=2, (3^3+8)=35 (this is not a simple n^3+1). Looking at differences: 35-2=33, 104-35=69, 209-104=105. Second differences: 69-33=36, 105-69=36. The next difference is 105+36=141. 209+141=350.

The sum of the arithmetic series is n/2 * (2*3 + (n-1)*3) = 3n(n+1)/2. We need 1.5n(n+1) > 1000, so n(n+1) > 666.67. Testing values, 25*26 = 650 (too small) and 26*27 = 702 (exceeds 1000).

The series consists of cubes: 12^3=1728, 14^3=2744, 16^3=4096, 18^3=5832, 20^3=8000, 22^3=10648. The next term is 24^3 = 13824.

The sequence is an AP with a = 3+4(1) = 7 and d = 4. Sum of n terms = n/2 * (2a + (n-1)d). For n=15: 15/2 * (2*7 + 14*4) = 15/2 * (14 + 56) = 15/2 * 70 = 15 * 35 = 525.

Differences: 4-0=4, 18-4=14, 48-18=30. Second differences: 14-4=10, 30-14=16. Third difference is 6. Next second difference = 16+6=22. Next first difference = 30+22=52. Next term = 48+52=100.