The sum of an infinite GP is given by S = a / (1 - r). Here, S = 10 and r = 1/3. Thus, 10 = a / (1 - 1/3) = a / (2/3) => a = 10 * (2/3) = 20.

The sum S_5 = 5/2 * (2a + 4d). Here, 50 = 5/2 * (2a + 8), solving gives a = 10.

The average of the first n terms is given by S_n/n. Here, S_5 = 50, so the average = 50/5 = 10.

Using the formula S_n = a(1 - r^n) / (1 - r), we have 63 = 3(1 - r^4) / (1 - r). Solving gives r = 2.

The sum S_n can be negative if the common ratio r is negative and the first term a is also negative.

As n approaches infinity and |r| < 1, r^n approaches 0, thus S_n approaches a/(1-r).

Using the formula S_n = a(1 - r^n) / (1 - r), we have 63 = 3(1 - r^4) / (1 - r). Solving gives r = 2.

Using the formula for the sum of a GP, S_n = a(1 - r^n) / (1 - r), we have 63 = 7(1 - 2^n) / (1 - 2). Solving gives n = 5.

The common difference can be found by calculating S_n - S_(n-1). S_n = 3n^2 + 2n and S_(n-1) = 3(n-1)^2 + 2(n-1). Simplifying gives the common difference as 6.

The sum of the first n terms S_n = n/2 * (2a + (n-1)d). By differentiating S_n with respect to n, we can find the common difference. The common difference is 3.

Let the first term be a. The sum of the first three terms is a + 2a + 4a = 7a. Setting 7a = 14 gives a = 2.

Let the first term be a. The sum of the first three terms is a + 2a + 4a = 7a. Setting 7a = 14 gives a = 2.

Let the first term be a. The sum of the first three terms is a + 3a + 9a = 13a. Setting 13a = 21 gives a = 21/13, which is not an option. Re-evaluating, if the common ratio is 3, the first term must be 7.

The numbers that satisfy both conditions are 4 and 8, as 4 + 8 = 12 and 4 * 8 = 32.

The second equation is a multiple of the first, indicating that both equations represent the same line.

The reciprocals are 1, 4, and 9. The common difference is 4 - 1 = 3.

The reciprocals of the terms are 1/3, 1/6, and 1/x. Since they form an arithmetic progression, we can set up the equation: 1/6 - 1/3 = 1/x - 1/6, solving gives x = 12.

The reciprocals are 1/4, 1/2, and 1/x. The common difference is 1/2 - 1/4 = 1/4, so 1/x = 1/2 + 1/4 = 3/4, thus x = 4/3.

Two lines are parallel if their slopes are equal, which occurs when a/d = b/e.

Both lines have the same slope (2) but different y-intercepts, indicating they are parallel.

Both lines have the same slope (2) but different y-intercepts, indicating they are parallel.

Lines with the same slope but different y-intercepts are parallel and will never intersect.

To find x, subtract 2 from both sides of the equation: x = 5 - 2, which gives x = 3.

To find x, subtract 3 from both sides: x = 7 - 3, which gives x = 4.

Substituting x and y gives us 2^(2+3) = 2^5 = 32.

Calculating, we find 2^3 + 3^2 = 8 + 9 = 17.

We have x = 8 and y = 4. Thus, x/y = 8/4 = 2.

The expression with a coefficient of 5 for x and a constant term of -3 is 5x - 3.

Since 125 can be expressed as 5^3, we have 5^(x+1) = 5^3, thus x + 1 = 3, leading to x = 2.

When a > 0 in a quadratic function, the graph opens upwards, indicating that the vertex is the minimum point.