# project-euler–problem 65

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

\( \sqrt{2} = 1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+?}}}} \)

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.

\( 1+\frac{1}{2} = 3/2 \)

\( 1+\frac{1}{2+\frac{1}{2}} = 7/5 \)

\( 1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}} = 17/12 \)

\( 1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}} = 41/29 \)

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 10th convergent is 1+4+5+7=17.

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

^{?}View Code RSPLUS

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e.expand <- function(n) { e0 <- 2 e <- c(1,2,1) add <- c(0,2,0) len <- floor(n/3)-1 x <- sapply(0:len, function(i) add * i) + e x <- as.vector(x) res <- c(e0, x) return(res) } get.fraction <- function(x) { ## 1/(x0 + a/x1) n <- length(x) x1 <- x[n-1] * x[n-2] + 1 a <- x[n-2] for (i in (n-2):1) { old.x1 <- x1; x1 <- x[i] * x1 + a a <- old.x1 } res <- list(numerator=x1, denumerator=a) return (res) } e <- e.expand(100) res <- get.fraction(e) numerator <- as.character(gmp::as.bigz(res$numerator)) s <- sum(as.numeric(unlist(strsplit(numerator, "")))) print(s) |

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