The "see and say" sequence is produced by first writing a 1,...

The "see and say" sequence is produced by first writing a $1,$ then it- erating the following procedure: look at the previous entry and say how many entries there are of each integer and write down what you just said. The first several terms of the "see and say" sequence are $1,11,21,1112,3112,211213,312213,212223, \ldots . .$ Comment on the ra- tionality (or irrationality) of the number whose decimal digits are ob- tained by concatenating the "see and say" sequence. $$ 0.1112111123112211213 \ldots $$

The "see and say" sequence is produced by first writing a $1,$ then it-
erating the following procedure: look at the previous entry and say
how many entries there are of each integer and write down what you
just said. The first several terms of the "see and say" sequence are
$1,11,21,1112,3112,211213,312213,212223, \ldots . .$ Comment on the ra-
tionality (or irrationality) of the number whose decimal digits are ob-
tained by concatenating the "see and say" sequence.
$$
0.1112111123112211213 \ldots
$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Rational vs. Irrational Numbers

Rational numbers are those that can be written as a fraction of two integers; their decimal expansion either terminates or eventually repeats. In contrast, irrational numbers cannot be expressed as a simple fraction and their decimals are non-terminating and non-repeating, highlighting a key difference in their structure.

Eventual Periodicity

Eventual periodicity is the property that, after some finite point, the digits of a decimal expansion repeat in a fixed cycle. This characteristic is a hallmark of rational numbers, and the lack of such periodicity in a decimal representation is used to establish the irrationality of a number.

Look-and-say Sequence

This is an iterative, combinatorial sequence where each term is produced by describing the previous term. In each step, the digits are read off in groups, and the count of each group followed by its digit is written down, forming the next term in the sequence.

Decimal Representation of a Number

This concept refers to the expression of a number in its base-10 form. In the given problem, a number is constructed by concatenating the terms of a sequence to form an infinite string of decimal digits, which is then interpreted as a real number.

Transcript

Please give Ace some feedback