Elephant sequence

Number Theory Level 2

An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes the result on the board. It stops after writing a number greater than one billion. How many distinct prime factors does the largest number on the board have?

2 4 1 3

2 solutions

A Former Brilliant Member
Jul 30, 2019

Assuming that the sequence is $\{a_n\}_{n=1}^{\infty}$ , where $a_1=1$ and $a_2=2$ , for $n>2$

$a_{n+1}=2 . (a_n+\frac{a_n}{2})=3.a_n$

therefore $a_n=2.3^{n-2}$ , for $n>2$

Ramesh Perumal
Jul 29, 2019

After listing a few terms, we notice that the numbers on the board have the following pattern: 1,2,2·3,2·3^2,2·3^3,.... One can show by induction using a geometric series that every term after the first will be twice a power of 3, where the exponent increases by 1 each time. Hence the largest number on the board will have exactly 2 prime factors.

Elephant sequence

2 solutions

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