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

The number of terms in the expansion of (x + y)^n is n + 1. Therefore, n + 1 = 15, which gives n = 14.

To solve for x, subtract 5 from both sides to get 3x = 15, then divide by 3 to find x = 5.

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.

Rearranging the sentences in the order of 2, 4, 3, 1 maintains a logical flow, building on the ideas presented before concluding with the initial statement.

The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = 7(1 - (1/2)^5) / (1 - 1/2) = 7(1 - 1/32) / (1/2) = 7 * 31/32 * 2 = 14.

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 sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = 10(1 - 0.5^5) / (1 - 0.5) = 10(1 - 0.03125) / 0.5 = 10 * 0.96875 / 0.5 = 19.375, which rounds to 20.

Let the common ratio be r. The sum is 10 + 10r + 10r^2 = 70. Simplifying gives r^2 + r - 6 = 0, which factors to (r - 2)(r + 3) = 0, thus r = 2.

The first three terms are 4, 4/3, and 4/9. Their sum is 4 + 4/3 + 4/9 = 4 + 1.33 + 0.44 = 5.77.

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 first term is 1, and the reciprocal of the second term in HP is 1 + 1 = 2. Therefore, 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 reciprocals are 1/3 and 1/6. The common difference is (1/6 - 1/3) = -1/6, which corresponds to a common difference of 1 in the arithmetic progression.

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 first term is 5, and the second term's reciprocal is 1/5 + 2 = 1/5 + 2/1 = 11/5. Therefore, the second term is 5/11, which is approximately 0.45.

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 first term is 5, the second term is 10, and the third term can be calculated as 1/(1/5 + 1/10) = 3.33. The sum is 5 + 10 + 3.33 = 18.33, which rounds to 20.