Math Puzzles and Number Sequences

Adding zero to any negative number preserves its negative value because zero is the additive identity. For example, -5 + 0 = -5. Zero doesn't change the sign or magnitude of any number. The result cannot be positive, zero, or infinity.

Division by zero is undefined in mathematics. When any non-zero number is divided by zero, the result is represented as a fraction with zero as the denominator, which is undefined. In limit contexts it may approach infinity, but technically the operation itself has no meaning in standard arithmetic.

Tetraneutron is a hypothetical particle consisting of four neutrons, which would have atomic number zero (no protons). It was proposed theoretically but never conclusively observed. The other elements have non-zero atomic numbers or different neutron configurations.

To find the last digit, we look at the units digit of each term. Numbers ending in 2 raised to any power have a cyclical pattern for the units digit: 2, 4, 8, 6 repeating every 4 powers. For 2002^2008, we check 2008 mod 4 = 0, so the units digit is 6. For 2008^2002, we check 2002 mod 4 = 2, so the units digit is 4. Adding: 6 + 4 = 10, which has a last digit of 0.

The pattern is: divide by 2 and add 1 alternately. 34/2 = 17, but the sequence shows 18, so it's 34-16=18, 18-8=10, 10-4=6, 6-2=4, 4-1=3. The subtracted values halve each time: 16, 8, 4, 2, 1.

The pattern is: multiply by 2, then subtract 1 alternately. 2×2=5? No, 2×2+1=5, 5×2-1=9, 9×2+1=19, 19×2-1=37, so 37×2+1=75. The +1, -1 alternates.

The sequence appears to follow the pattern: 1^1+1=2, 2^2=4, 3^2=9, 5^3=125. Each term relates to powers of increasing integers. Option A (125) fits as 5^3, continuing this numerical progression.

The sequence shows increasing differences: -2 to -2 (difference 0), -2 to 0 (difference 2), 0 to 4 (difference 4). The differences increase by 2 each time (0, 2, 4), so the next difference is 6, making 4+6=10.

Each term is the sum of all previous terms: 0, then 0+1=1, then 0+1=1, then 0+1+1=2, then 0+1+1+2=4. The next term is 0+1+1+2+4=8.

The sequence shows increasing differences: -2 to -2 (difference 0), -2 to 0 (difference 2), 0 to 4 (difference 4). The differences increase by 2 each time (0, 2, 4), so the next difference is 6, making 4+6=10.

Speed upstream = Speed in still water - Speed of stream. Given: Upstream speed = 6 km/h, Stream speed = 2 km/h. Therefore: Speed in still water = Upstream speed + Stream speed = 6 + 2 = 8 km/h.

Let the two-digit number be 10x + y (where x is tens digit, y is units digit). The reversed number is 10y + x. Their difference: (10x + y) - (10y + x) = 9x - 9y = 9(x-y) = 36, so x-y = 4. The difference between the digits is 4.