Question

Find the number of common terms in the following sequences (each of which is an A.P) which are less than 10,000
1st sequence 37,103,169.....
2nd sequence 17,82,147.....

Answer 1

The first sequence has a common difference of D1 = 66.

The second sequence has a common difference of D2 = 65.

LCM of these two is 4290.

There is already a difference of 20 in the starting terms of the two APs.

And on each successive term the difference increases 1.

So in the first term the difference is 20, the next is 21, the further is 22 and so on.

When this difference hits the lower of the two common differences then we have our first common term.

Thus if 1st difference is 20 then nth difference is 65.

Thus n is 46.

Thus the 46th term in both APs is

37 + (46-1)66 = 3007
17 + (46-1)65 = 3007

Since the LCM is greater than the first common term, we can look for greater common terms

Thus next term would be
3007 + 4290 = 7297

But above 7297 we will cross the limit of 10000. Thus there are only two common terms... which are 3007 and 7297.