Consecutive integer pyramid

Question

The first 4 rows of a pyramid of consecutive integers is 1, then 2 3 4, then 5 6 7 8 9, then 10 11 12 13 14 15 16 and so on.

Every other row has 2 more entries than the row directly above it.

What is the value of the 2013th entry in the 2013th row?

score 1 · Answer 1 · answered Aug 20 '17 at 18:34

1

Notice that the last digit of each row is $n^2$ (this can pretty easily be proved with induction, and is the same as the proof that the sum of the first $n$ consecutive odd numbers is $n^2$), where $n$ is the row. So, it is $2012^2 + 2013$.

answered Aug 20 '17 at 18:34

Duncan Ramage

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score 0 · Answer 2 · answered Aug 20 '17 at 18:36

Guide:

In the first line you have one number.

In the first two lines you have $1+(1+2)$ number.

In the first third line you have $1+(1+2)+(1+2+2)$ number.

Hence the number of entries in the first $n$ lines is an arithmetic sum.

How many number are there in the first $2012$ lines?

Consecutive integer pyramid

2 Answers2