Even More Algebra 2 Help?

Question

Once again im back. im seriously considering a weekly installment of this. this time i am honestly lost and i have tried every possible thing i know to try and solve this problem. it goes exactly like this.

The counting numbers are listed in the triangular array shown below. in what row does the number 2007 appear(forgive me because im not sure if i can put the numbers in a triangle.)

1
2 3 4
5 6 7 8 9
10 11 12 13 14 15 16
17 18 19 20 21 22 23 24 25


please help me if you can and show all work that is necessary. thanks...

Answer 1

If you let n = the row number, the number at the end of the row is n^2. If you find the perfect square just larger than 2007, the square root of that will be the number of the row. The square root of 2007 is 44.7 (calculator), so the next perfect square would be that of 45. That's the row number.

Answer 2

Notice these things: the first row ends in 1^2 = 1, the second ends in 2^2=4, the third ends in 3^2 = 9, and so on.

The largest square less that 2007 is 1936 = 44^2. Therefore the 44th row ends in 1936. Then the 71st number in the 45th row will be 2007.

Cute problem. Now test yourself with a few easy questions like: Where did the number 47 occur? Where did 85 occur?

Answer 3

You already have some good solutions, but since I worked on this I'll answer it too.
We have a another sequence: 1, 4, 9, 16, . . . which give the numbers used by the end of the first, second, third, etc. rows. Of course, these are the integers squared in sequence. What we want is the first integer n such that n^2 greater than 2007, which is 45.
What course is this for? [e-mail]

Answer 4

Let
row(n) = # of elements in the nth row, so
row(1)=1
row(2)=3
row(3)=5
or
row(n) = 2n - 1
Let
total(n) = # of numbers up to row n, so
total(1)=1
total(2)=1 + 3 = 4
total(3) = 4 + 5 = 9
or
total(n) = Sum (2k-1) for k =1 to n
= 2(Sum of 1 to n ) - n
= n(n+1) - n
= n^2
In other words, 1 to nth row has n^2 elements

Now 44^2 = 1936
and 45^2 = 2025
since 2007 is between these numbers,
it is on row#45 - - - Ans