The term is given by C(n, 3) * a^2 * x^3. Setting C(n, 3) * a^2 = 15 gives n = 6.

In the term x^4y^2, the sum of the exponents (4 + 2) must equal n, hence n = 6.

The sum of the exponents in the term x^4y^3 is 4 + 3 = 7, hence n must be 7.

The sum of the exterior angles of any polygon is 360 degrees. Therefore, the number of sides can be calculated as 360 / 30 = 12.

The exterior angle is equal to the sum of the two opposite interior angles. If the exterior angle is 120 degrees, the smallest interior angle can be calculated as 180 - 120 = 60 degrees.

The equation y = k/x represents a family of hyperbolas.

Removing the first sentence would eliminate important context, weakening the argument presented.

The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = x(1 - (1/2)^5) / (1 - 1/2) = x(1 - 1/32) / (1/2) = x(31/32) * 2 = x(62/32).

The nth term of a GP is given by a * r^(n-1). Here, the 6th term = x * 3^(6-1) = x * 3^5.

The nth term of a geometric series is given by a_n = ar^(n-1). Here, a = 4, r = 2, n = 7. So, a_7 = 4 * 2^(7-1) = 4 * 64 = 256.

The first three terms are 4, 12, and 36. Their product = 4 * 12 * 36 = 1728.

The 6th term is given by 7 * 3^(6-1) = 7 * 243 = 1701.

The nth term of a GP is given by a * r^(n-1). Thus, the 4th term is x * y^(4-1) = xy^3.

The first term is 1, and the second term's reciprocal will be 1 + 2 = 3. Therefore, the second term is 1/3.

The first term is 1, and the second term's reciprocal will be 1 + 1 = 2, so the second term is 1/2.

The reciprocals are 1 and 2, which have a common difference of 1.

The reciprocals of the terms are 1, 3, and 1/x. The common difference is 2, so 1/x = 3 + 2 = 5, thus x = 1/5.

The first term is 4, and the reciprocal is 1/4. The second term's reciprocal will be 1/4 + 2 = 9/4, so the second term is 4/9.

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 fourth term's reciprocal will be 1/10 - 1/10 = 1/25, hence the fourth term is 25.

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 number of terms n can be calculated using the formula: n = (last - first) / difference + 1. Here, n = (50 - 10) / 5 + 1 = 9.

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.

Using the formula for the nth term, a + (n-1)d = 7 + 14*2 = 7 + 28 = 35.

Using the formula for the last term, l = a + (n-1)d, we have 37 = 7 + (16-1)d. Solving gives d = 2.

The nth term of an arithmetic sequence is given by a + (n-1)d. Here, a = 5, d = 3, n = 10. So, 5 + (10-1)3 = 32.

In an arithmetic series, the last term can be expressed as a + (n-1)d. Here, 48 = 12 + (10-1)d. Thus, 48 - 12 = 9d, giving d = 4.

a_n = a + (n-1)d = 5 + (15-1) * 3 = 5 + 42 = 47.

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