Using the following formula for a triangular number, verify that 1225 and 41 616 are triangular numbers. [Hint verify that there is an n for each number that satisfies the equation]
e.g., t5 = 15 =(5+1)/2
tn = n(n+1)/2
so, should I plug in 1225 and 41,616 and work the equation out? Somehow, I'm just reall confused by this problem. Could someone please be so kind as to help me?
no, find n for each. I will do one.
41,616= n(n+1)/2
83,232= n^2+n
put it in binomial form, and solve
n^2+n-83232=0
n= (-1+-sqrt 332929)/2= -1/2 +-288.5
n= 288, or n=-289
Now put them in the formula and verify. Normally, we restrict n to positive values, but your problem did not.
Now do the other number.
oh, okay! I'll try it out; thanks so much!
sorry, but I don't get where you're getting "n= (-1+-sqrt 332929)/2= -1/2 +-288.5" could you please explain?
1) I made an error in the next to last line.
n=0,180,360....the 1 was a typo.
Now, how did I get n. It came from the binomial equation for the quadratic form
if
ax^2+bx + c=0, then
x= (-b+-sqrt(b^2-4ac))/2a
oh... okay, thanks for the correction and the explanation
Yes, you're on the right track! To verify if a number is a triangular number using the given formula tn = n(n+1)/2, you need to find a value for n that satisfies the equation.
Let's start with 1225 as an example:
Step 1: Plug in 1225 into the formula: tn = 1225
1225 = n(n + 1)/2
Step 2: Multiply both sides of the equation by 2 to eliminate the division:
2 * 1225 = n(n + 1)
Step 3: Simplify the equation:
2450 = n^2 + n
Step 4: Rearrange the equation to obtain a quadratic equation equal to zero:
n^2 + n - 2450 = 0
Step 5: Factorize or use the quadratic formula to solve for n.
If the equation can be factorized, it would be:
(n - x)(n - y) = 0
If you cannot factorize it, you can use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 1, and c = -2450.
Step 6: Solve for n:
Use the factored form or the quadratic formula to find the values of n that satisfy the equation. If n is an integer, then the number is a triangular number. If n is not an integer, then the number is not a triangular number.
Perform the same steps for the number 41,616 to determine if it is a triangular number.
I hope this helps! Let me know if you have any further questions.