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MATH / For a positive integer n, the n^(th) triangular number is T(n)=(n(n+1))/(2). For example, T(3)=(3(3+1))/(2)=(3(4))/(2)=6, so the third triangular number is 6. Determine the smalles…

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For a positive integer nn, the nthn^{th} triangular number is T(n)=n(n+1)2.T(n)=\dfrac{n(n+1)}{2}. For example, T(3)=3(3+1)2=3(4)2=6T(3) = \frac{3(3+1)}{2}= \frac{3(4)}{2}=6, so the third triangular number is 6. Determine the smallest integer b>2011b>2011 such that T(b+1)−T(b)=T(x)T(b+1)-T(b)=T(x) for some positive integer xx.
Plain-text mathematical notation (without MathML)
For a positive integer n, the n^(th) triangular number is T(n)=(n(n+1))/(2).

For example, T(3)=(3(3+1))/(2)=(3(4))/(2)=6, so the third triangular number is 6.

Determine the smallest integer b>2011 such that T(b+1)−T(b)=T(x) for some positive integer x.
Original LaTeX notation
For a positive integer $n$, the $n^{th}$ triangular number is $T(n)=\dfrac{n(n+1)}{2}.$

For example, $T(3) = \frac{3(3+1)}{2}= \frac{3(4)}{2}=6$, so the third triangular number is 6.

Determine the smallest integer $b>2011$ such that $T(b+1)-T(b)=T(x)$ for some positive integer $x$.

level

Level 5

type

Algebra

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Source and history

task_id
a414324e-ee10-5b76-85c7-ba8c12ed2aed
task_key
algebra--test--a414324e-ee10-5b76-85c7-ba8c12ed2aed
task_revision_id
1
config / split
algebra / test

Official source

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