The first term in the arithmetic progression is 1/5, and the common difference is 2. Therefore, the second term in the harmonic progression is 1/(1/5 + 2) = 1/(2.2) = 5/11.

The reciprocals are 1/5 and 1/10. The common difference is -1/10. The third term's reciprocal will be 1/10 - 1/10 = 1/15, so the third term is 15.

The reciprocals are 1/5 and 1/10. The common difference is -1/10. The fourth term's reciprocal will be 1/10 - 1/10 = 1/25, hence the fourth term is 25.

The last term can be expressed as a + (n-1)d. Here, 48 = 12 + 9d. Solving gives d = 4.

Using the formula for the nth term, a + (n-1)d = 7 + (8-1)(-2) = 7 - 14 = -7.

The reciprocals are 1, 2, and 3. The common difference is 2 - 1 = 1.

The reciprocals are 1, 2, and 3, which are in arithmetic progression. The next term in the sequence of reciprocals is 4, so the fourth term is 1/4.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression, hence 1/a + 1/b = 1/c.

A function that is symmetric about the y-axis satisfies the property f(x) = f(-x) for all x.

A function is symmetric about the y-axis if it is an even function, which satisfies the condition f(x) = f(-x).

To find the x-intercept, set y = 0. Solving gives x = 2, so the intersection point is (2, 0).

Setting y to 0 in the equation gives x = 2, so the intersection with the x-axis is (2, 0).

'a' represents the first term of the harmonic progression in the formula for the nth term.

Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.

Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

According to Vieta's formulas, the sum of the roots p and q is -b/a and the product is c/a.

For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0, leading to k = 4.

According to Vieta's formulas, the sum of the roots p + q is equal to -(-5) = 5.

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.

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.

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.

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

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

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

The coefficient 'a' in a quadratic function determines whether the parabola opens upwards (a > 0) or downwards (a < 0).

The function has critical points where the first derivative is zero, which can be analyzed to find both local maxima and minima.

The nth term of a GP is given by a * r^(n-1). Here, a = 3, r = 2, and n = 5. Thus, the 5th term = 3 * 2^(5-1) = 3 * 16 = 48.