Consider the following series: 1,2,3,4,5,10,20,40....which starts as an arithmetic series?
...but after the first five terms becomes a geometric series. Prove that any positive integer can be written as a sum of distinct numbers from the series.
I know how to do the base case
all the numbers from 1-9 can be formed from the 1st five terms.
All multiples of 10 can be formed from the geometric terms (basically just a scaled binary sequence).
With that, all the gaps between the tens can be filled with the units.
Hey steve, can you show how to solve that using strong induction?
To prove that any positive integer can be written as a sum of distinct numbers from the given series, we can use strong induction.
We will assume that the statement holds true for all positive integers up to a certain value, and then show that it holds for the next integer.
Base Case: We can start with the number 1, which is the first term of the series. So the base case is proven.
Inductive Hypothesis: Assume that all positive integers up to some positive integer value 'n' can be written as a sum of distinct numbers from the series.
Inductive Step: We need to prove that 'n+1' can also be written as a sum of distinct numbers from the series.
Since the terms after the 5th term form a geometric series, let's call the common ratio 'r'. In this case, 'r' is equal to 2 because each term is twice the previous term.
Now, let's consider any positive integer 'n'. If 'n' is one of the terms in the series up to the 5th term, then 'n' can be written as a sum of itself, as each term in the series is distinct.
If 'n' is not one of the terms up to the 5th term, we can write 'n' as a sum of numbers from the previous terms in the series. We can use the concept of binary representation to achieve this.
For example, let's consider the number '6'. In binary form, '6' can be represented as '110'. Each digit in the binary representation represents if we include a particular term or not. In this case, '1' in the first digit represents including the term '4', '1' in the second digit represents including the term '2', and '0' in the third digit represents not including the term '1'.
So, '6' can be written as '4 + 2'. Similarly, any positive integer can be represented as a sum of distinct numbers from the series using binary notation.
Now, since we assumed that the statement holds true for all positive integers up to 'n', we know that '1', '2', '3', ..., 'n' can all be written as a sum of distinct numbers from the series.
Using the binary notation, we can express 'n+1' as a sum of distinct numbers from the series by representing 'n+1' in binary form and including the corresponding terms from the series.
Therefore, by the principle of strong induction, we have proven that any positive integer can be written as a sum of distinct numbers from the given series.
To prove that any positive integer can be written as a sum of distinct numbers from the given series, we will use mathematical induction. This involves proving a base case, and then showing that if the statement holds for some number, it also holds for the next number.
Base Case (n=1):
The given series starts with the number 1. Since we need to show that any positive integer can be written as a sum of distinct numbers from the series, the base case is already satisfied.
Assumption (Assuming the statement holds for some number, k):
Let's assume that any positive integer up to k can be written as a sum of distinct numbers from the series.
Inductive Step (Proving that the statement holds for the next number, k+1):
We need to prove that any positive integer up to (k+1) can be written as a sum of distinct numbers from the series.
Case 1: If (k+1) is already present in the series
In this case, (k+1) itself is a distinct number from the series, and we can write (k+1) as a sum of just (k+1).
Case 2: If (k+1) is not present in the series
Since the series starting from the 6th term forms a geometric series, we can express any number in that series as a product of powers of 2. The sequence from the 6th term is: 10, 20, 40, 80, ...
We can express any number in this sequence as 10 * 2^(n-5), where n is the position of the number in the sequence.
We can also see that every number in this sequence can be expressed as the sum of distinct numbers from the series that starts as an arithmetic series (1, 2, 3, 4, 5) followed by a geometric series (10, 20, 40, 80, ...).
Since we assumed that any positive integer up to k can be written as a sum of distinct numbers from the series, we can express (k+1) as (k+1) = 10 * 2^(n-5) + a, where a is a sum of distinct numbers from the series (1, 2, 3, 4, 5).
Therefore, we have shown that any positive integer up to (k+1) can be written as a sum of distinct numbers from the series.
By the principle of mathematical induction, we have proved that any positive integer can be written as a sum of distinct numbers from the given series.