# triangular numbers: In what row is the number 1000?

#### soccerisgreat

##### New member
The positive integers are written in a triangular array. In what row is the number 1000? So, if I want the number at the end of the 10th row, I take 10*11/2=55. But how do I find what row a number is in?

#### galactus

##### Super Moderator
Staff member
You're thinking on the right lines. 55 occurs on the 10th row. Because

$$\displaystyle \L\\\sum_{k=1}^{10}k=55$$

You can use $$\displaystyle \L\\\frac{n(n+1)}{2}=1000$$

Solve for n. That should give you a good look at what row it's in.

#### skeeter

##### Senior Member
each row ends with a member of the triangular number sequence
1, 3, 6, 10, 15, ...

the nth term of this sequence is n(n+1)/2

to find the row where 1000 resides ...

set n(n+1)/2 = 1000

n(n+1) = 2000

n should be close to the square root of 2000 ... approx 44.7

check the n-values close to 44.7 using the nth term above ...

44*45/2 = 990

45*46/2 = 1035

1000 lies in the 45th row.

#### TchrWill

##### Full Member
soccerisgreat said:
The positive integers are written in a triangular array. In what row is the number 1000? So, if I want the number at the end of the 10th row, I take 10*11/2=55. But how do I find what row a number is in?
The first numbers in the triangular rows are 1, 2, 4, 7, 11,...etc. and are defined by N = (n^2 - n + 2)/2

Setting this expression equal to 1000 yields an n = 45.2 meaning that the number 1000 is in the 45th row.