- #1
oszust001
- 10
- 0
How can I show that:
[tex]\sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2^{n}} [/tex]
for every natural numbers
[tex]\sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2^{n}} [/tex]
for every natural numbers
Version of AtomSeven is good.AtomSeven said:The identity is wrong, it should be
[tex]
\sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2 n}
[/tex]
An equation in natural numbers is a mathematical statement that shows the relationship between two or more natural numbers. It consists of two sides, the left side and the right side, with an equal sign in between. The goal is to find the value of the unknown number(s) that make the equation true.
To solve an equation in natural numbers, you must isolate the unknown number(s) on one side of the equal sign by performing the same operation on both sides of the equation. The order of operations (PEMDAS) should be followed, and the solution must be a natural number, which is a positive whole number (excluding 0).
Yes, an equation in natural numbers can have more than one solution. This is because there can be multiple combinations of natural numbers that make the equation true. For example, the equation 3x + 2 = 8 has two solutions, x = 2 and x = 6.
The only difference between an equation in natural numbers and an equation in whole numbers is that natural numbers do not include 0, while whole numbers include both 0 and positive integers. Therefore, an equation in whole numbers can have 0 as a solution, while an equation in natural numbers cannot.
Equations in natural numbers are used in various real-life situations, such as calculating the cost of items when shopping, determining the number of people in a group based on the total number of items they have, and solving problems in fields like engineering, economics, and physics. They are also used in creating patterns and sequences in mathematics and computer science.