Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, then its 4$$^{th}$$ term is:

We are given an arithmetic progression (A.P.) with all positive integer terms. The second term is 12, and the sum of the first nine terms is greater than 200 and less than 220. We need to find the fourth term.

Let the first term be $$a$$ and the common difference be $$d$$. Since the second term is 12, we have:

$$a + d = 12 \quad \text{(Equation 1)}$$

The sum of the first nine terms of an A.P. is given by:

$$S_9 = \frac{9}{2} \times [2a + (9-1)d] = \frac{9}{2} \times [2a + 8d] = \frac{9}{2} \times 2(a + 4d) = 9(a + 4d)$$

Given that this sum is between 200 and 220:

$$200 < 9(a + 4d) < 220$$

Dividing the entire inequality by 9:

$$\frac{200}{9} < a + 4d < \frac{220}{9}$$

Calculating the decimals:

$$\frac{200}{9} \approx 22.222 \quad \text{and} \quad \frac{220}{9} \approx 24.444$$

So:

$$22.222 < a + 4d < 24.444$$

Since $$a$$ and $$d$$ are integers (as terms are positive integers), $$a + 4d$$ must be an integer. The only integers between 22.222 and 24.444 are 23 and 24. Therefore, we have two cases:

Case 1: $$a + 4d = 23$$

Case 2: $$a + 4d = 24$$

We also have Equation 1: $$a + d = 12$$.

Solving Case 1:

Subtract Equation 1 from $$a + 4d = 23$$:

$$(a + 4d) - (a + d) = 23 - 12$$

$$a + 4d - a - d = 11$$

$$3d = 11$$

$$d = \frac{11}{3} \approx 3.666$$

But $$d$$ must be an integer since terms are integers, so this case is invalid.

Solving Case 2:

Subtract Equation 1 from $$a + 4d = 24$$:

$$(a + 4d) - (a + d) = 24 - 12$$

$$a + 4d - a - d = 12$$

$$3d = 12$$

$$d = 4$$

Substitute $$d = 4$$ into Equation 1:

$$a + 4 = 12$$

$$a = 8$$

So the first term $$a = 8$$ and common difference $$d = 4$$.

The fourth term is:

$$T_4 = a + (4-1)d = 8 + 3 \times 4 = 8 + 12 = 20$$

Verification:

The first nine terms are: 8, 12, 16, 20, 24, 28, 32, 36, 40.

Sum: $$8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 = 216$$

$$200 < 216 < 220$$ is true, and all terms are positive integers.

The options are:

A. 8

B. 24

C. 20

D. 16

Hence, the correct answer is Option C.