The nth term of an AP is given by a + (n-1)d. Here, a = 5, d = 3, and n = 10. So, the 10th term = 5 + (10-1) * 3 = 5 + 27 = 32.

Using the nth term formula, a + (n-1)d = 7 + 5*(-2) = 7 - 10 = -3.

Using the formula S_n = n/2 * (2a + (n-1)d), we have 100 = 10/2 * (2a + 9*2). Solving gives a = 10.

Using the formula S_n = n/2 * (2a + (n-1)d), we can substitute n = 10 and d = 5 to find a = 20.

Using the sum formula S_n = n/2 * (2a + (n-1)d), we have 50 = 5/2 * (10 + 4d). Solving gives d = 7.

Using the sum formula S_n = n/2 * (2a + (n-1)d), we have 50 = 5/2 * (2a + 8). Solving gives a = 10.

The sum of the first n terms is given by S_n = n/2 * (2a + (n-1)d). Here, 50 = 5/2 * (2a + 8). Solving gives a = 10.

The decimal number 255 is represented as FF in hexadecimal.

The hexadecimal number 'A3' converts to decimal as follows: A (10)*16^1 + 3*16^0 = 160 + 3 = 163.

The hexadecimal number 1A converts to decimal as 1*16^1 + 10*16^0 = 26.

The word 'MATH' has 4 distinct letters. The number of arrangements is 4! = 24.

Each gift can go to any of the 4 children, so the total ways = 4^3 = 64.

The number of ways to choose 3 students from 10 is given by 10C3 = 120.

The number of ways to choose 3 students from 8 is given by 8C3 = 56.

Treat the 2 specific books as one unit. Then, we have 3 units to arrange: (2 books together) + (2 other books) = 3! * 2! = 12.

The number of arrangements of 4 different books is 4! = 24.

The number of ways to choose 4 fruits from 10 is given by 10C4 = 210.

Each gift can go to any of the 3 children, so the total ways are 3^4 = 81.

Each gift can go to any of the 3 children, so the total ways are 3^4 = 81.

The word 'SCHOOL' has 6 letters with 'O' repeating 2 times. The arrangements are 6! / 2! = 360.

Using the formula A = P(1 + r)^n, we set 3P = P(1 + 0.1)^n. Solving gives n ≈ 12 years.

Using the formula A = P(1 + r)^n, we set A = 3P and solve for n, which gives approximately 12 years.

A fallacy is an error in reasoning that renders an argument invalid, often leading to incorrect conclusions.

Topology is the branch of mathematics that studies the properties of space that are preserved under continuous transformations.

Linear algebra is important as it deals with vector spaces and linear mappings, which are foundational in various applications across mathematics and science.

Topology is the branch of mathematics that studies the properties of space that are preserved under continuous transformations.

The multiplicative inverse of 3 mod 11 is 4, since 3 * 4 ≡ 12 ≡ 1 (mod 11).

The multiplicative inverse of 3 mod 11 is 4, since (3 * 4) mod 11 = 12 mod 11 = 1.

All operations (addition, subtraction, multiplication) are valid in modular arithmetic.

For any integer a, a mod 2 will always yield either 0 or 1, depending on whether a is even or odd.