The challenge for you is to solve for x.

Here’s a hint: think of triangular numbers!

Once again, this is a good moment to pause the article and give the problem a go yourself. When you’re ready, keep reading for the solution. And if you come up with your own approach, feel free to share it in the comments — I’d love to see how you tackled it.

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Solution

First, notice that the number in the denominator of each fraction is a triangular number.

By rewriting the expression in descending order, we can add the two expressions together and derive a general formula for the nth triangular number.

Notice we have n sets of n + 1 in our summation of the two triangular numbers.

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We will also need partial fraction decomposition to solve our problem.

We have contained a decomposed form of our fraction. We will now apply the above techniques to solve for x.

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Factoring out x, we have

We then use the formula of triangular numbers to rewrite each individual fraction.

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Substituting these expressions into the denominators, we have

which gives us,

By manipulating the 1 into the same form as other fractions, we get

Factoring the 2 from every numerator gives us

We will now use our partial fraction decomposition formula derived earlier

Notice the terms in the middle cancel out and we are left with the first and the last term

Therefore, the answer is

Bonus: Can you find the general expression for which the sum equals n?

And that’s our answer. How amazing. Hey, don’t forget to clap👏 the article as a token of appreciation. Thank you 🦁