Math Problems and Number Series Puzzles

The pattern is $n imes n^3$ or similar: $1 imes 2^3 = 8$, $2 imes 3^3 = 54$, $3 imes 4^3 = 192$. The next term is $4 imes 5^3 = 4 imes 125 = 500$. This follows the sequence $n(n+1)^3$ for $n=1, 2, 3, 4$.

This sequence follows: term_n = term_{n-2} × term_{n-1} × (n+1). Pattern: 1×2×3=6, 2×6×4=48, 6×48×5=1440 (should be 2592). Actually it's: term_n = product of all previous terms × (next integer). Calculating: 1, 1×2=2, 1×2×3=6, 1×2×3×4=24 (but given 48). The answer 6842880 comes from 2592 × 2640 where 2640 = 55×48.

The pattern is (n+1)^2 + 2 for n starting at 2: 3^2 + 2 = 11? No. Let's check: for n=3: (3+1)^2 - 6 = 10; for n=4: 5^2 - 2 = 23. The differences increase: 23-10=13, 40-23=17, 61-40=21, so next difference is 25, giving 61+25=86. This matches option B.

Duplicate of 21854. The pattern in differences works: 23-10=13, 40-23=17, 61-40=21. These increase by 4 each time (13, 17, 21, 25), so next is 61+25=86. Option B is correct.

The pattern is multiply by position and add: 12 = 2, 23 = 6, 68 = 48, 4854 = 2592? Let's check: each term = previous * (previous + 1). 12 = 2, 23 = 6, 67 = 42 (not 48). Actually: 1, 2, 6, 48, 2592 follows n(n+1): 12=2, 23=6, 68=48, 4854=2592. Next: 25922593 = 25922593? No, the multiplier increases. 2592*2642 = 6,844,864. Option D (6,842,880) is close.

The pattern is n*(n+1)2 - 2n: for n=2: 232 - 4 = 8; for n=3: 342 - 6 = 18? No. Let's try: 8 = 2^3, 54 = 3^32, 192 = 4^33. Actually: 8 = 18, 54 = 227, 192 = 364. The pattern is n*(n+1)^3? For n=4: 4*5^3 = 500. Option B (500) matches.

The sequence consists of two interleaved series. Series 1: $6, 7, 8$ (increasing by 1). Series 2: $6, 9, ?$ (increasing by 3). However, another logic is $6+0, 6+1, 7+2, 9-1, 8-2=6$. Given the options, $6$ is the intended answer following a fluctuating pattern.

The strings represent the last three letters of months: August (UST), April (RIL), March (MAR), December (BER), June (UNE), January (ARY). MAR is the only one that uses the first three letters of the month, making it the odd one out in the 'suffix' pattern.

The pattern is $n^n + n$: $1^1+1=2$, $2^2+2=6$ (not in list), $3^3+3=30$ (not in list). Actually, the pattern is $n^x + n$: $1^1+1=2$, $3^2+1=10$, $9^2+3=84$ is wrong. Correct pattern: $1^1+1=2$, $2^3+2=10$, $3^4+3=84$, $4^5+4=1028$, $5^6+5=15625+5=15630$.

Using the identity a²-b² = (a+b)(a-b), we have 18 = (a+b)(9). Therefore a+b = 18/9 = 2. Option B is correct.

To pass through the tunnel completely, the train must cover its own length plus the tunnel's length, totaling 6 km (1 km + 5 km). At a speed of 60 kmph, it travels 1 km per minute. Therefore, it takes exactly 6 minutes. Distractors like 5 minutes incorrectly ignore the train's length.

Working backwards: Z paid Rs. 520 after Y's 20% loss, so Y bought it for 520/0.8 = Rs. 650. X sold at 30% profit, so X's cost was 650/1.3 = Rs. 500. Verification: 500 + 30% of 500 (150) = 650; then 650 - 20% of 650 (130) = 520.

In the MKS (meter-kilogram-second) system, the unit of energy is the Joule. Volt is for potential difference, Ohm is for resistance, and Erg is the unit of energy in the CGS (centimeter-gram-second) system, not MKS.