Soln:

If we have 2 AP’s and are trying to find common terms, then the common terms also form a AP with a CD of LCM of the 2 common differences.

The first sequence is of the form 17 + 4a

And the second sequence is of the form 16 + 5b

By observation we see that 21 is the first term of the new sequence and the common difference is 20 ( LCM of 4 and 5)

Hence the sequence will be 21 + 20c

Now substitute values of c from the answer options and check that the last term should not cross 417.

Hence when we sub c=19 we see that the value is lesser than 417 and when we sub c=20 the value is 421 which is greater than 417.

Hence the number of terms will be 19 + first term = 20 terms or option ( c)