Given the series 1,2,3,4,5,10,20,40,80,..... which starts as an arithmetic series for the first five terms and then follows geometric series. It can be proved by strong induction that any positive integer can be written as a sum of distinct numbers of this series.

Has to be solved using strong induction. How do you do this??

To prove that any positive integer can be written as a sum of distinct numbers from the given series, we can use strong induction.

Step 1: Base cases
First, we need to prove that the statement holds for the smallest possible values. In this case, let's consider the positive integers 1, 2, 3, 4, and 5. From the given series, we can see that each of these integers can be written as a sum of distinct terms:
1 = 1
2 = 2
3 = 1 + 2
4 = 4
5 = 1 + 4

So, the base cases are true.

Step 2: Inductive Hypothesis
Next, we assume that the statement holds for all positive integers up to a certain positive integer 'n' and try to prove it for the next positive integer 'n+1'.

Step 3: Inductive Step
Let's assume that 'k' is a positive integer such that any positive integer up to 'k' can be written as a sum of distinct terms from the series.
We want to prove that the statement also holds for 'k+1'.

- First, consider the case when 'k+1' is already a term in the series. In this case, 'k+1' can be written as a sum of only itself, which is a distinct term from the series. So, the statement holds.

- Next, consider the case when 'k+1' is not a term in the series. We know that 'k' can be written as a sum of distinct terms from the series. Let's denote this sum as S. Then, 'k+1' can be written as (S + 1). Since 'k+1' is not a term from the series, we can express it as (1 + a) where 'a' is a combination of terms from the series. By substituting 'S' back, we can rewrite (S + 1) as (1 + a + S), which is a sum of distinct terms from the series. So, the statement holds in this case as well.

Therefore, by strong induction, we have shown that for any positive integer 'n', it can be written as a sum of distinct numbers from the given series.

Note: The provided series is a combination of an arithmetic series (1, 2, 3, 4, 5) and a geometric series (10, 20, 40, 80, ...). This combination allows for any positive integer to be expressed as a sum of distinct terms.

To prove that any positive integer can be written as a sum of distinct numbers from the given series using strong induction, you would typically follow these steps:

1. Base Case: Show that the statement holds for the smallest possible value. In this case, we need to show that the statement is true for 1. Here, 1 can be expressed as 1, which is one of the terms in the series.

2. Induction Hypothesis: Assume that the statement is true for all positive integers up to some integer k. This assumption is called the induction hypothesis.

3. Induction Step: Use the induction hypothesis to prove that the statement holds for k+1.

In this case, let's assume that the statement holds for all positive integers up to some integer k.

Now, we need to prove that the statement also holds for k+1.

Consider the following cases:

Case 1: k+1 is one of the terms in the series (e.g., k+1 = 2, 3, 4, 5, 10, 20, 40, 80, ...).

In this case, k+1 can be expressed as k+1 itself, which is one of the terms in the series. Therefore, the statement holds.

Case 2: k+1 is not one of the terms in the series.

In this case, we know that k can be expressed as the sum of distinct terms from the series.

Let's consider the largest term in the series that is less than or equal to k. This term, let's call it "m," is less than k+1.

Now, we can express k+1 as the sum of m and a positive integer x such that x ≤ k.

Since we assumed that the statement holds for all positive integers up to k, we can express x as the sum of distinct terms from the series.

Therefore, k+1 can be expressed as the sum of distinct terms from the series.

By completing these steps, we have shown that if the statement holds for some positive integer k, it also holds for k+1. Therefore, by the principle of strong induction, the statement holds for all positive integers.