#### Problem

The square root of 2 can be written as an infinite continued fraction.

The infinite continued fraction can be written, ?2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, ?23 = [4;(1,3,1,8)].

It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for ?2.

Hence the sequence of the first ten convergents for ?2 are:

1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, …

What is most surprising is that the important mathematical constant,

e = [2; 1,2,1, 1,4,1, 1,6,1 , … , 1,2k,1, …].

The first ten terms in the sequence of convergents for e are:

2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, …

The sum of digits in the numerator of the 10^{th}convergent is 1+4+5+7=17.

Find the sum of digits in the numerator of the 100^{th}convergent of the continued fraction for e.

#### Solution

If you look at the convergents of *?2* and the numerators in the convergents of *e*, you’ll see a pattern emerging:

1 + 2 * 1 = 3

2 + 3 * 2 = 8

3 + 8 * 1 = 11

8 + 11 * 1 = 19

11 + 19 * 4 = 87

If you look at the sequence of numerators in the convergents of *e* ( 2, 3, 8, 11, … ) and the sequence of numbers in the convergents of *?2 *( 1, 2, 1, 1, 4, 1, … ), given the current numerator ( *n *) and the previous numerator ( *n-1* ) in the sequence and the corresponding number in the convergents of *?2* ( *i* )the next numerator ( *n+1* ) can be calculated using the formula:

( n+1) = ( n-1 ) + n * i

Once we have this formula to work with, the rest is simple, the solution runs in 7 milliseconds on my machine.

**Yan Cui**

I’m an **AWS Serverless Hero** and the author of **Production-Ready Serverless**. I have run production workload at scale in AWS for nearly 10 years and I have been an architect or principal engineer with a variety of industries ranging from banking, e-commerce, sports streaming to mobile gaming. I currently work as an independent consultant focused on AWS and serverless.

Further reading

Here is a complete list of all my posts on serverless and AWS Lambda. In the meantime, here are a few of my most popular blog posts.

- Lambda optimization tip – enable HTTP keep-alive
- You are thinking about serverless costs all wrong
- Many faced threats to Serverless security
- We can do better than percentile latencies
- I’m afraid you’re thinking about AWS Lambda cold starts all wrong
- Yubl’s road to Serverless
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- AWS Lambda – compare coldstart time with different languages, memory and code sizes
- Guys, we’re doing pagination wrong