How many different ways can a number $n \in \mathbb{N} $ be expressed as a sum of any number of positive numbers when order matters?
My solution:
Since I know, that $n$ can be represented as a sum of $k$ positive integers ${{n-1}\choose{k-1}}$ different ways, and $n$ can be represented as a sum of at most $n$ positive integers (the case where they are all $1$), then we can present $n$ as a sum of of any number of integers $\sum\limits_{k=1}^{n} {{n-1}\choose{k-1}}$ different ways.
Your solution is correct. $n \in \mathbb{N}$ can be expressed as the sum of a sequence of natural numbers in $$\sum\limits_{k=1}^{n}{n-1 \choose k-1} = 2^{n-1}$$ different ways. This is called a composition of an integer $n$. See Wikipedia Article for more information.
Imagine that you are climbing an $n$-step staircase, and can take any size of steps you wish. Let $W(n)$ be the number of ways to do it. We have $W(1)=1$.
We now express $W(n+1)$ in terms of $W(n)$. Either our first step is $1$, in which case there are $W(n)$ ways to finish the climb. Or else the first step is $a\gt 1$. In that case, by making the first step $a-1$, we get a way of climbing an $n$-step staircase, and all ways of climbing such a staircase can be obtained in this way.
Thus $W(n+1)=W(n)+W(n)=2W(n)$. It follows that $W(n)=2^{n-1}$ for all $n$.
(Apply deductive/inductive reasoning) A nice number is a number that can be expressed as the sum of a string of two or more consecutive positive integers a) Determine which numbers from 50 to 70 inclusive are nice. Answer The numbers that are inclusive here are the n odd numbers from 1 to n = n2 just a matter of subtracting the squares. In this case, there are 5 odd numbers that are less than 10 like 10/5 which sums up to 52 = 25 or 102/4 going by this number or count input solution 1225 – 625 = 600 or (4900 – 2500)/4 = 600. The sum of the first n odd numbers is n2 considering that this is 502 = 2500 the first is square 1 tile making an arrangement of 2×2 the additional tile is contained on the right side which can be extended to an arrangement of 3×3 with an additional 2 tile on both sides with 1 at the corner right side thus 3+3+1 = 7 resulting to a series of 1+3+5+7+9+1. Let U denote the entire set of numbers. Let P and B denote the set of numbers from 50 to 70 inclusive are nice. Thus: |U| = 50 |B| = 70 |P| = 25 |P ∩ B| = 10
b) Show that numbers that are powers of 2 are nice Let the first set create the series of numbers that are of power 2 A={… 1, 2, 4, 8, 16, 0.0625, 0.125, 0.25, 0.5 … } which is the same as A={… 2^0, 2^1, 2^2, 2^3, 2^4, 2^(–4), 2^(–3), 2^(–2), 2^(–1) …} This value can be written as A={2^n | n is N} where N represents the set of values that are infinite and countable with countable infinity. c) Show that positive integers which are not powers of 2 are not nice Suppose that 800×800= 640000 with the square of 800 ends containing four zeros with a positive integer square cannot end with four non zero equal digits with a known perfect square of 1111,2222,3333,…….9999 the values should be 5555 or 6666 or 9999 or 1111 or 4444 considering that the one’s digits squared number cannot be 2,3,7, or 8. d) Given a nice number, can you determine how many different ways it can be expressed as a sum of positive integers? n=3(1+2,2+1),n=4(1+3,3+1,2+2),n=5(1+4,4+1,2+3,3+2) n=2(1+1),n=3(1+2,2+1),n=4(1+3,3+1,2+2),n=5(1+4,4+1,2+3,3+2)n=2(1+1), Thus, the integral solution is x1+x2=nandx1,x2>0x1+x2=nandx1,x2>0 which is the same way used in finding the values of (n−2) without a 0 for either x1 or x2 which are the identical balls that fall under different distinct bins that places a cross divide of (n – 2) taking the values of divider across the values of (n – 2) being identical thus becoming.
