Why is (1,3,6,10,\ldots) not an arithmetic progression?

Because consecutive differences are not equal

An arithmetic progression is a sequence in which the difference between every pair of consecutive terms is the same constant. The terms may increase or decrease, but equal spacing is the essential requirement. In the sequence \(1,3,6,10,\ldots\), the first difference is \(3-1=2\), the next is \(6-3=3\), and the next is \(10-6=4\). Since these differences are not equal, the sequence is not an arithmetic progression.

Therefore, option C is correct. The fact that the terms increase does not by itself make a sequence an AP; for example, increasing differences can still occur. The first term being 1 is irrelevant, and an AP does not need a displayed final term because it may continue indefinitely. The decisive test is equality of consecutive differences.

This is a Class 10 Mathematics question. Chapter: Arithmetic Progressions (AP). Topic: Introduction to APs and common difference..