The Riddle of the Recursive Sequence

Ive been fascinated by riddles recently, specifically math related ones. Given that is one of my interests and strong suits, I decided to make one myself. Try to solve it! (Answer at the bottom of the page)

Imagine you are a mathematician tasked with solving a puzzle presented at a mathematics conference. The problem was submitted anonymously, and nobody knows its orgins. It is said that the answer could unlock a groundbreaking new discovery about number sequences.

You find a note on your desk that reads:

The Riddle:
“I begin with a humble number,
A single 1, the simplest form.
But each step I take is guided,
By what came before, my shape reborn.”

“Each term I take from all the past,
Divided by the distance cast.
To find me, trace my path ahead,
To where ten steps have been led.”

“I grow as I step along,
Each one building where the others belong.
From start to end, a hidden sum,
Will reveal the answer when you’re done.”



The Question:

What is the sum of the first 10 terms of this recursive sequence?

STOP HERE! ANSWERS AHEAD!



Understanding the riddle:

The riddle describes a recursive sequence where each term is calculated based on the previous term. Specifically

  • The first term starts with 1.

  • Each subsequent term is based on the previous terms, but divided by a factor that depends on its position in the sequence.

    • The sequence begins with 1, so \(S_1 = 1\). The riddle hints that each term is influenced by all of the previous terms, and division is involved. For a term \(S_n\), the relationship between terms seems to involve a sum of previous terms divided by the difference in their position. (i.e., the difference in their index.)



    From the clues in the riddle, the can interperet the recursive formula as \(S_n = \sum_{k=1}^{n-1} \frac{S_k}{n - k}\)
    This means to find each term, we have to sum the previous terms divided by the difference in their indices. For example,
    \(S_1\) is the starting term, and is given as 1.
    \(S_2\) depends only on \(S_1\), and is simply \(\frac{S_1}{2-1}\).
    \(S_3\) depends on \(S_1\) and \(S_2\), and is \(\frac{S_1}{2-1}+\frac{S_2}{3-2}\).
    You get it..

    Calculating the terms

    We now calculate each term based on the formula:

    \(S_1 = 1\) (Starting point)

    \(S_2\):

    For \(S_2\), we use the formula and only consider \(S_1\):

    \[ S_2 = \frac{S_1}{2 - 1} = \frac{1}{1} = 1 \]

    \(S_3\):

    For \(S_3\), we sum \(S_1\) and \(S_2\) using the formula:

    \[ S_3 = \frac{S_1}{3 - 1} + \frac{S_2}{3 - 2} = \frac{1}{2} + \frac{1}{1} = 0.5 + 1 = 1.5 \]

    \(S_4\):

    For \(S_4\), we add the terms for \(S_1\), \(S_2\), and \(S_3\):

    \[ S_4 = \frac{S_1}{4 - 1} + \frac{S_2}{4 - 2} + \frac{S_3}{4 - 3} = \frac{1}{3} + \frac{1}{2} + \frac{1.5}{1} = 0.3333 + 0.5 + 1.5 = 2.3333 \]

    Continuing the process for \(S_5\), \(S_6\), \(S_7\), etc.:

    You continue this process for each term, adding the fractions based on the recursive formula and updating the terms step by step.

    When you have all 10 of the terms, you sum them to find the golden sum- the riddle's answer.

    \[ S_{\text{sum}} = S_1 + S_2 + S_3 + \ldots + S_{10} \]

    After calculating all the terms, we get:

    \[ S_{\text{sum}} \approx 100.60 \]


    There is our answer! Wasn't that just so easy! What makes this problem challenging is because of the recursive nature of the sequence. Each new term is derived from all of the previous terms, and the division by the difference in the positions introduces a complex twist and makes the terms grow differently than in a simple arithmetic or geometric progression. The pattern is hard to predict without solving the terms because of this-- each term depends on a growing sum of the previous ones divided by their index difference.

    So basically, the golden sum is approximately 100.60-- the sum of the first 10 terms of the recursive sequence.

    Hope this riddle make you scratch your head a bit, it was fun to make!