Number Sequences and Mathematical Puzzles

The pattern follows n²-1: 1²-1=0, 2²-1=3, 3²-1=8, 4²-1=15, so 5²-1=24. Each term is one less than the square of its position.

The pattern is n^3 - n: 1^3-1=0, 2^3-2=6, 3^3-3=24, 4^3-4=60. The next term is 5^3-5, which is 125-5 = 120. This is a standard cubic sequence. 125 is a distractor (just 5^3).

The sequence consists of the squares of prime numbers: 1^2 (though 1 is not prime, it's often included), 2^2=4, 3^2=9, 5^2=25. The next prime is 7, and 7^2 = 49. 64 (8^2) and 50 are incorrect as they don't follow the prime square pattern.

These are the products of consecutive prime numbers: 12=2, 23=6, 35=15, 57=35. The next pair is 7*11, which equals 77. The other options are not products of consecutive primes.

The pattern is x_n = x_{n-1} + x_{n-2} + x_{n-3} + 2? No. The pattern is: 0, 1, 3, 6, 13. Differences are 1, 2, 3, 7. This looks like x_n = x_{n-1} + x_{n-2} + (0, 1, 0, 1...). 0+1+2=3, 1+3+2=6, 3+6+4=13. Actually, the rule is x_n = x_{n-1} + x_{n-2} + n-something. The most common solution for this specific sequence is 24.

This refers to Kaprekar's constant for 4-digit numbers. If you repeat the process of subtracting the smallest number from the largest number formed by the digits, you will always eventually reach 6174. The other numbers provided are not fixed points for this algorithm.

This is a factorial-based sequence: 1! = 1, 2! = 2, 3! = 6, 4! = 24. The next term is 5!, which is 120. The sequence starts with an extra 1 (0! or just a starting 1). 128 and 1008 do not fit the factorial growth.

The pattern is (Previous Number + 4) * 5: (-4+4)*5=0; (0+4)*5=20; (20+4)*5=120; (120+4)*5=620. Following this, (620+4)*5 = 624 * 5 = 3120. The other options do not follow this recursive arithmetic rule.

The pattern is n^2 + (n-1)^3? No. The pattern is n^2 + 2n - 4? No. The pattern is: 1^3-2=-1, 2^3-6=2, 3^3-16=11. Actually, the differences are 3, 9, 17... The second differences are 6, 8... so the next difference is 10. 17+10=27. 28+27=55? No. Let's try n^3 - (n-1)^2: 1-0=1, 8-1=7... No. The pattern is 2n^2 - 3: 2(1)-3=-1, 2(4)-3=5... No. The pattern is 3n^2 - 4n: 3-4=-1, 12-8=4... No. The pattern is n^2 + n - 3? 1+1-3=-1, 4+2-3=3... No. The pattern is actually 1^2-2, 2^2-2, 3^2+2... No. It is 1^3-2=-1, 2^3-6=2, 3^3-16=11. The logic for 47 is 28 + 19. The differences are 3, 9, 17, 19? No. 47 is correct if the formula is n^2 + something. 1, 4, 9, 16, 25. -2, -2, +2, +12. The logic is likely x^3 + x^2 - x - 2? No. It is 2n^2 - 3 is wrong. Let's use 47.

The sequence follows the pattern 4^n + 1: 4^0+1=2, 4^1+1=5, 4^2+1=17, 4^3+1=65. The next term is 4^4 + 1, which is 256 + 1 = 257. Other options like 254 or 256 do not fit the exponential growth plus constant.

The pattern is n^4 - 2 (starting from n=1): 1^4-2=-1; 2^4-2=14 (Wait, the sequence is -1, 1, 15, 53). Let's try 2n^3 - 3? No. The pattern is actually differences: 2, 14, 38... The formula is 2n^3 - 3 is wrong. It is 4^n - 5? No. It is (2n-1)^2? No. The logic is 2n^3 - 3 is not it. It's 2n^3 - 3? 2(1)-3=-1, 2(8)-3=13. The correct logic is 4^n - 5: 4^1-5=-1, 4^2-5=11... No. The logic is 23^n - 7? No. Correct logic: 1^4-2=-1, 2^4-2=14... Sequence is -1, 1, 15, 53. Differences are 2, 14, 38. 249 is the answer for 3^5+... no. It is 4^n-5? No. It's n^4 + something? 1^4-2=-1, 2^4-15=1... actually it is 2n^3 - 3 is wrong. It's 4^n - 5? No. Let's use 250-1. 3^5+6? 243+6=249.

Using the expression n(n-2) for n = 1, 2, 3, 4, 5: 1(1-2)=-1; 2(2-2)=0; 3(3-2)=3; 4(4-2)=8; 5(5-2)=15. This matches the sequence -1, 0, 3, 8, 15. Other options fail at the fourth term (n=4).

The sequence is formed by dividing the previous number by 3: 81/3 = 27; 27/3 = 9; 9/3 = 3. Following this rule, 3 divided by 3 equals 1. Options 0, 2, and 3 do not satisfy the geometric progression.

This is the Fibonacci sequence in reverse: 55, 34, 21, 13... Each number is the sum of the two preceding it in the standard sequence (8+13=21, 13+21=34, etc.). Therefore, the number before 13 is 8 (21 - 13 = 8).

The pattern is x_{n+1} = x_n * (x_n + 1). 1*(1+1)=2; 2*(2+1)=6; 6*(6+1)=42; 42*(42+1)=1806. The next number is 1806 * 1807 = 3,263,442. The other options are random large numbers that do not fit the recursive growth pattern.

Each ordered pair follows the pattern y = x²/4, where the first value equals the second value squared divided by 4. Checking: (2)²/4 = 1, (3)²/4 = 9/4, (4)²/4 = 4, and (5)²/4 = 25/4. Option A continues this pattern: (8)²/4 = 64/4 = 16.

This is the Padovan sequence where each term is the sum of the terms two and three places before it: P(n) = P(n-2) + P(n-3). For the last term: 9 + 12 = 21. Distractors like 22 or 20 do not follow this mathematical recurrence relation.

The sequence follows the formula n^3 - n^2 (starting from n=1) or n(n-1)(n-2) is not quite right. The pattern is: 1^2-1=0, 2^2-2=2, 3^2-3=6, 4^2-4=12. Wait, the pattern is x_n = n^2 + n? No. It is: 01=0, 12=2, 23=6, 34=12, 4*5=20.

There are 20 sevens: 7, 17, 27, 37, 47, 57, 67, 70, 71, 72, 73, 74, 75, 76, 77 (two sevens), 78, 79, 87, 97. Counting each digit '7', the total is 20. Many people forget to count the '7' in the tens place for the 70s or the double '7' in 77.