The nth term of the series can be found using T_n = S_n - S_(n-1). First, calculate S_10 and S_9: S_10 = 3(10^2) + 2(10) = 320, S_9 = 3(9^2) + 2(9) = 273. Thus, T_10 = S_10 - S_9 = 320 - 273 = 47.

The common difference can be found by calculating S_n - S_(n-1). This gives the common difference as 5.

The 10th term can be found using T_n = S_n - S_(n-1). Here, S_10 = 5(10) + 3 = 53 and S_9 = 5(9) + 3 = 48. Thus, T_10 = 53 - 48 = 5.

The 4th term can be found using T_n = S_n - S_(n-1). Here, S_4 = 5(4) + 3 = 23 and S_3 = 5(3) + 3 = 18. Thus, T_4 = 23 - 18 = 5.

The 5th term can be found using a_n = S_n - S_(n-1). Calculate S_5 and S_4, then find a_5 = S_5 - S_4 = (5(5^2) + 3(5)) - (5(4^2) + 3(4)) = 38.

'a' represents the first term of the arithmetic series.

'd' represents the common difference in an arithmetic series.

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 sum of the exterior angles of any polygon is always 360 degrees.

From Vieta's formulas, p = -sum of roots = -5 and q = product of roots = 6.

The sum of the roots is -p = 8, hence p = -8.

The sum of the roots is given by -(-3)/1 = 3. Thus, k = 3 - 3 = 0.

The sum of the roots is given by -b/a = 3. Here, -(-3)/1 = 3, so p can be any value.

The sum of the roots is given by -(-6)/1 = 6. The product of the roots is k, and since the sum is correct, k can be any value.

Let the integers be n, n+1, n+2. Then n + (n+1) + (n+2) = 72. Solving gives 3n + 3 = 72, so n = 24.

Let the integers be x, x+1, and x+2. Their sum is x + (x + 1) + (x + 2) = 3x + 3 = 72. Solving gives 3x = 69, so x = 23.

Angles that sum up to 90 degrees are called complementary angles.

Two angles whose sum is 90 degrees are called complementary angles.

All pairs listed are multiples of 7 that sum to 56.

The numbers are 4 and 8, as 4 + 8 = 12 and 4 × 8 = 32.

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

Let the other number be x. Then, 9 + x = 15. Thus, x = 15 - 9 = 6.

Let the two numbers be x and y. We have x + y = 15 and xy = 50. Solving these equations gives the larger number as 10.

Let the other number be x. Then, x + 12 = 20. So, x = 20 - 12 = 8.

Let the numbers be x and y. We have x + y = 20 and xy = 96. Solving these equations gives x = 12 and y = 8.

The two numbers are 12 and 8, as they satisfy both the sum (12 + 8 = 20) and the product (12 * 8 = 96).

The sum of the remainders is 4 + 4 = 8, and 8 divided by 7 gives a remainder of 1.

If one number is 20 (a multiple of 5), the other number would be 10, making it a valid option.