There is a triangular number that has 55 dots in its shape. Which one is it? Write your answer as a number. The triangular number that has 55 dots in its shape is the
-th triangular number.

9514 1404 393

  10

Step-by-step explanation:

The n-th triangular number is given by ...

  t(n) = n(n+1)/2

We went to find n when t(n) = 55.

  55 = n(n+1)/2

  110 = n(n+1)

Adding 1/4 completes the square.

  110.25 = (n +0.5)^2

  √110.25 = n+0.5 . . . . . we are interested in the positive value of n

  n = 10.5 -0.5 = 10

The triangular number that has 55 dots in its shape is the 10-th number.

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Additional comment

Here, we have gone to the trouble to formally complete the square to find the value of n. You may realize that it isn't really necessary to go to that trouble.

A reasonable estimate of the value of n is possible by considering that the product n(n+1) is a little more than n², so the value of n will be a little less than √110 ≈ 10.49. The nearest integer is 10, which is the answer we're looking for.

step-by-step explanation:

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