Concept:It is a question of common AP. 1st series. :- 17, 21, 25 2nd series. :- 16, 21, 26 • 1st terms of common AP = 1st common visible no. So,. a = 21• common difference of common AP = LCM of D1&D2 Here, D1= 4. &. D2 = 5 D = LCM of (4,5) = 20. Now, new series will be, 21, 41, 61....... Sum of 1st 200 terms = =200/2(2*21 + (200-1)20) =100(42+199*20) = 100*4022 =402200.Hence, correct answer is (B).

Arithmetic Progressions

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. The general form of an arithmetic progression is given by:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term and 'd' is the common difference.

Finding the Terms in Common

To find the numbers that appear in both arithmetic progressions, we need to compare the terms of the two progressions. Let's denote the terms of the first progression as a1, a2, a3, ... and the terms of the second progression as b1, b2, b3, ....

Comparing the two progressions, we have:
a1 = 17, a2 = 21, a3 = 25, ...
b1 = 16, b2 = 21, b3 = 26, ...

From this, we can see that 21 appears in both progressions.

Finding the Common Difference

Now that we know the common term, we can find the common difference by subtracting consecutive terms. In this case, the common difference is the same for both progressions.

For the first progression:
a2 - a1 = 21 - 17 = 4
a3 - a2 = 25 - 21 = 4

For the second progression:
b2 - b1 = 21 - 16 = 5
b3 - b2 = 26 - 21 = 5

Thus, the common difference is 4 or 5.

Finding the Number of Terms

To find the number of terms that appear in both progressions, we need to determine the position of the last common term. Let's denote the position of the terms in the first progression as n1 and the position of the terms in the second progression as n2.

We can find the position of the last common term by using the formula:
last term = first term + (n - 1) * common difference

For the first progression:
a + (n1 - 1) * 4 = last common term

For the second progression:
a + (n2 - 1) * 5 = last common term

Since the last common term is the same in both progressions, we can set these two equations equal to each other and solve for n:
17 + (n1 - 1) * 4 = 16 + (n2 - 1) * 5

Simplifying the equation, we get:
4n1 - 4 = 5n2 - 5
4n1 - 5n2 = 1

Finding the Sum of Terms

To find the sum of the terms that appear in both progressions, we can use the formula for the sum of an arithmetic progression:

sum = (number of terms / 2) * (first term + last term)

In this case, the number of terms is n1 for the first progression and n2 for the second progression. However, since n1 and n2 are not given, we cannot directly calculate the sum.

However, the question asks for the sum of the first 200 terms appearing in both progressions, which implies that the 200th term is the last common term. Therefore,