DIRECTIONS for question 4: Type in your answer in the input box provided below the question.If the first four terms of an arithmetic progression are p, p + 2q, 3p + q and 30 respectively, find the value of the 2016th term of the progression.

The first step in solving this problem is to understand the properties of an arithmetic progression (AP). In an AP, the difference between any two successive terms is constant. This difference is called the common difference.

Given the first four terms of the AP as p, p + 2q, 3p + q, and 30 respectively, we can find the common difference by subtracting the first term from the second term, and the second term from the third term.

So, the common difference (d) is (p + 2q) - p = 2q and (3p + q) - (p + 2q) = 2p - q.

Setting these two equations equal to each other gives us 2q = 2p - q. Solving for p gives us p = 3q.

Substituting p = 3q into the third term of the AP gives us 3p + q = 30. Substituting p = 3q into this equation gives us 9q + q = 30, or 10q = 30. Solving for q gives us q = 3.

Substituting q = 3 into p = 3q gives us p = 9.

Now that we have the first term (a) as 9 and the common difference (d) as 6, we can find the 2016th term of the AP using the formula for the nth term of an AP, which is a + (n - 1)d.

Substituting a = 9, d = 6, and n = 2016 into this formula gives us the 2016th term as 9 + (2016 - 1) * 6 = 12099.