How many square numbers are there among the numbers 1 to 1000?
How many triangular numbers are there among the numbers 1 to 1000?
perfect squares are of the form n^2
so you want n^2 < 1000
n < √1000 < appr 31.6...
31^2 = 961
32^2 =1024
So there are 31 perfect squares less than 1000
Triangular numbers are 1,3,6,10,15,...
(think of racking up billiard balls in its triangle and adding the balls.
The first row has 1 ball, after 2 rows you have 3 balls, after 3 rows you have 6 balls, etc. That is why they are called triangular numbers.)
These numbers are produced by n(n+1)/2
we want:
n(n+1)/2 < 1000
n^2 + n < 2000
n^2 + n - 2000 < 0
If we had n^2 + n - 2000 = 0
n = appr 44.2.. or n is a negative
if n = 44, 44(45)/2 = 990 , if n = 45, we go greater than 1000
So there are 44 triangular numbers less than 1000
n^2 <= 1000
n <= √1000 = 31.6
The nth triangular number is n(n+1)/2
So, you need
n(n+1)/2 <= 1000
n <= 44.2
To determine the number of square numbers among the numbers 1 to 1000, we need to find the perfect squares within that range.
First, we start by finding the square root of 1000. The square root of 1000 is approximately 31.62. Since we are looking for the perfect squares up to 1000, we can conclude that the perfect squares within the range are 1^2, 2^2, 3^2, up to 31^2. So, there are 31 square numbers among the numbers 1 to 1000.
Now let's move on to finding the number of triangular numbers within the same range.
Triangular numbers are obtained by summing consecutive natural numbers. For example, 1 + 2 = 3, 1 + 2 + 3 = 6, and so on.
To find the triangular numbers up to 1000, we need to check for a pattern in the sequence of triangular numbers. We can observe that the nth triangular number is given by the formula T(n) = n * (n + 1) / 2.
If we solve T(n) = 1000, we find n ≈ 44.1. This means that the largest triangular number within the range of 1 to 1000 will be T(44) = 44 * (44 + 1) / 2 = 990.
Therefore, there are 44 triangular numbers among the numbers 1 to 1000.