Strange numbers

Can you guess the next number of the sequence 4, 12, 24, 40, 60?

You might have already tried some algebra. Might have done some combinations of AP and/or GP. But did it fit into any of it? If yes, then ping me, I would love to know it.

I came to this sequence from a problem statement which states that there are a few activities, the first takes k time, the second takes 2k time, the third takes 3k time, and soo on. In this case, the value of k is 4 and hence the sequence represents the time to finish each activity.

So on the basis of this knowledge let’s reverse engineer this sequence. We’ll calculate the difference between the consecutive numbers and write them as follows.

Now you would clearly see an AP sequence forming with a common difference of 4.

These sequences are called Quadratic Sequences because their general term is a polynomial of degree 2.

Let’s now learn how to derive the general term of these sequences. We start with writing the nth term as an² + bn + c. Now if we can find the values of a, b and c respectively then our job will be done.

Now we need to know some formulas.
1. 2a = second difference
2. 3a+b = difference of first two terms
3. a+b+c = first term

Using them,
2a = 4
=> a = 2

3a+b = 12 – 4
=> 3(2) + b = 8
=> b = 8-6
=> b = 2

a+b+c = 4
=> 2 + 2 + c = 4
=> c = 0

And hence the nth term of the above sequence will be 2n² + 2n + 0.

I hope you learned something new from this article :)