Number Sequences and Basic Math Formulas

The value of Pi (π), the ratio of a circle's circumference to its diameter, is approximately 22/7. This rational approximation (approximately 3.142857) is commonly used in calculations. The exact value is an irrational number approximately equal to 3.14159. The question has a typo ('PIE' instead of 'Pi') but the intended answer is 22/7.

The expansion of (A + B)³ is A³ + 3A²B + 3AB² + B³, which can be factored as A³ + B³ + 3AB(A + B). This is a standard algebraic identity. Option A incorrectly uses 2AB instead of 3AB, while options C and D have incorrect structures.

The algebraic expansion of (A-B)^3 is A^3 - B^3 - 3AB(A-B). This can also be written as A^3 - 3A^2B + 3AB^2 - B^3. Option 407822 correctly represents the factored form of the subtraction terms.

The perimeter of a rectangle is the sum of all four sides, which equals 2 × (Length + Breadth). This is because opposite sides are equal, so you add Length + Breadth and multiply by 2. Option A (L × B) would give area, not perimeter.

The sequence increases by 11 each time: 12+11=23, 23+11=34, 34+11=45. Therefore, 45+11=56. The other options are incorrect as they do not maintain the constant difference of 11.

The pattern alternates between multiplying by 5 and subtracting 7: 7 × 5 = 35, 35 - 7 = 28, 28 × 5 = 140, 140 - 7 = 133. The next step continues with × 5: 133 × 5 = 665. This is not simple addition or subtraction.

The logic is: (AB) followed by (A+B) as a string. 32=6 and 3+2=5 (Wait, the logic is AB and A+B reversed or similar). Actually, it is (AB) and (A+B): 32=6, 3+2=5 (46 is 22 and 3+3?). Correct logic: First digit is (A+B-1) and second is (AB). No, it is (A+B+1) and (AB)? For 47: 47=28, 4+7=11. 148 fits if the logic is (A+B+3) and (AB)? No, the logic is (A+B) and (AB) with specific offsets. 4*7=28, 4+7=11. 11+3=14. Result 148.

The differences between consecutive terms increase by 2: 9-6=3, 14-9=5, 21-14=7, 30-21=9, 41-30=11, 54-41=13. The missing number is 41, which fits perfectly in the sequence (30 + 11 = 41, then 41 + 13 = 54)

The pattern works in groups of three: (5+6)=11, then 116=66. For the next set: (9+7)=16, then 167=112. The logic is (a+b)=c, then c*b=d. 112 is the result of multiplying the sum (16) by the second number in its set (7).

The sequence consists of pairs where the second number is the square or cube of the first, or simply powers. 7, 8, 7^2(49), 8^2(64), 7^3(343), 8^3(512). The missing number is 7 cubed, which is 343.

Each term (from the third onward) is the sum of the two preceding terms: 1+1=2, 1+2=4, 2+4=7, 4+7=11. The next term is 7+11=16. This Fibonacci-like sequence continues indefinitely.

The differences between consecutive terms are increasing even numbers: 35-33=2, 33-29=4, 29-23=6, 23-15=8. The next difference should be 10, so 15-10=5. The other options do not match this arithmetic progression of differences.

The series follows the formula n^2 - 1 starting from n=3: 3^2-1=8, 4^2-1=15, 5^2-1=24, 6^2-1=35, 7^2-1=48, 8^2-1=63. The next term is 9^2-1=80. Alternatively, the differences are increasing odd numbers: 7, 9, 11, 13, 15, so 63+17=80.

The pattern is differences: 4-2=2, 10-4=6, 24-10=14, 54-24=30. These differences follow the formula 2^(n) - 2: 2, 6, 14, 30, 62. So 54 + 62 = 116. Alternatively, each term is (previous × 2) + 2: 2×2+2=6, 4×2+2=10, 10×2+2=22(not 24)... let me verify. Actually: 2, 4, 10, 24, 54 follows: a_n = 2^(n) + 2^(n-1) - 2? Better to see differences: 2, 6, 14, 30 - each is 2×(previous) + 2. So next difference is 62, giving 54+62=116.

The pattern alternates between subtracting 2 and adding 1: 19-2=17, 17+1=18, 18-2=16. To continue the pattern, the next step is to add 1: 16+1=17. The distractors 12, 14, and 15 do not follow this specific alternating logic.

The pattern alternates between dividing by 2 and adding 4: 96/2=48, 48+4=52, 52/2=26, 26+4=30. The next step is to divide by 2: 30/2=15. Distractors like 14 or 16 do not fit this arithmetic-division cycle.

This sequence has two interleaved patterns: odd positions increase by 2 (20, 22, 24, 26) and even positions decrease by 1 (10, 9, 8). The seventh position (odd) continues 24 + 2 = 26. The eighth position (even) continues 8 - 1 = 7.

This is an alternating series. The first series (1st, 3rd, 5th, 7th terms) increases by 1: 1, 2, 3, 4. The second series (2nd, 4th, 6th terms) decreases by 1: 7, 6, 5. The next term in the second series would be 4.

The series follows a multiplication pattern where each term is multiplied by an increasing even number: 22=4, 43=12, 124=48. Following this logic, the next multiplier is 5, and 485=240. Other options do not follow this geometric progression.

The sequence consists of two alternating series. The first series (positions 1, 3, 5, 7) is squares of even numbers: 2^2=4, 4^2=16, 6^2=36, 8^2=64. The second series (positions 2, 4, 6) is squares of even numbers in descending order: 14^2=196, 12^2=144, 10^2=100. The next term is the 8th term, belonging to the second series: 8^2=64.