If a > 0 in a quadratic function, the graph opens upwards, indicating that it has a minimum point.

'a' determines the direction of the parabola's opening; if 'a' is positive, it opens upwards, and if negative, it opens downwards.

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

'a' determines the direction of the parabola; if 'a' is positive, it opens upwards, and if negative, it opens downwards.

'a' determines the direction of the parabola; if 'a' is positive, it opens upwards, and if negative, it opens downwards.

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

If f(a) = f(b) for a ≠ b, it indicates that the function is not one-to-one, which means it does not pass the horizontal line test.

Let the common ratio be r. The 5th term is given by ar^4 = 64. Thus, 4r^4 = 64 => r^4 = 16 => r = 2.

Let the first term be a. The 3rd term is a * r^2 = a * 3^2 = 9a. Setting 9a = 27 gives a = 3.

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.

The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 4/32 = 0.125.

The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_4 = 5(1 - 0.5^4) / (1 - 0.5) = 5(1 - 0.0625) / 0.5 = 5 * 0.9375 / 0.5 = 9.375.

The last term can be expressed as a * r^(n-1). Here, 80 = 5 * r^(4-1) = 5 * r^3. Thus, r^3 = 16, giving r = 2.

Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.

The sum of the first n terms of a GP is given by S_n = a(1 - r^n)/(1 - r) for r ≠ 1.

The 3rd term of a GP is given by a * r^(n-1). Here, it is x * y^(3-1) = xy^2.

Let the first term be a and the common ratio be r. Then, 3rd term = ar^2 = 27 and 5th term = ar^4 = 243. Dividing gives r^2 = 9, so r = 3. Substituting back gives a = 3.

The 6th term is given by 10 * (0.5)^(6-1) = 10 * (0.5)^5 = 10 * 0.03125 = 0.3125.

The 4th term is given by 2 * (-2)^(4-1) = 2 * (-8) = -16.

The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 0.125, which is 0.25.

The first four terms are 5, 2.5, 1.25, and 0.625. Their sum is 5 + 2.5 + 1.25 + 0.625 = 9.375.

Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.

The 6th term is given by 7 * (1/2)^(6-1) = 7 * (1/32) = 0.4375.

The nth term of a GP is given by a * r^(n-1). Thus, the 6th term = x * y^(6-1) = xy^5.

The first term is 1, the second term is 1/2, and the third term can be calculated as 1/(1 + 1/2) = 2/3. The sum is 1 + 1/2 + 2/3 = 2.

The reciprocals are 1 and 2, which are in arithmetic progression with a common difference of 1/2.

The reciprocals are 1 and 2, which are in arithmetic progression. The third term's reciprocal is 3, so the third term is 1/3.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 3 are 1/2 and 1/3. The common difference is 1/3 - 1/2 = -1/6. The third term's reciprocal will be 1/3 - 1/6 = 1/6, so the third term is 6.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of 2 and 4 are 1/2 and 1/4. The common difference is -1/4. Therefore, the third term's reciprocal is 1/4 - 1/4 = 0, which means the third term is 1.

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. The reciprocals of the first two terms are 1/2 and 3/4. The common difference is 1/4, so the reciprocal of the third term is 1/2 + 1/4 = 3/4. Therefore, the third term is 1/(3/4) = 4/3.