The last term in the sum is "2na+ b".rooski said:Homework Statement
give an inductive proof for all n >= 1
(2a+b) + (4a+b) + ... + (2na + b) = n(an + a + b)
The Attempt at a Solution
In order to start the method of inductoin i have to prove that f(1) holds true.
I am confused by the ... in this equation though.
If n = 1, then would the equation be (2a+b)+(4a+b)+(2a+b) = n(an + a + b) ?
Proof by induction is a mathematical technique used to prove that a statement holds true for all natural numbers. It works by showing that the statement is true for a starting value, usually 0 or 1, and then proving that if the statement is true for a particular value, it must also be true for the next value. This process is repeated until the statement is proven to be true for all natural numbers.
In the context of this equation, proof by induction is used to show that the equation holds true for all natural numbers. The base case is usually proven by plugging in the starting value, and then the inductive step is used to show that if the equation holds true for a particular value, it must also hold true for the next value. This process is repeated until the equation is proven to hold true for all natural numbers.
The starting value for this equation is typically 1, as indicated by the "n = 1" in the equation. However, it is important to note that the starting value can vary depending on the context of the problem.
The inductive step in this equation involves showing that if the equation holds true for a particular value of n, it must also hold true for the next value, or n+1. This is typically done by plugging in n+1 for n in the equation and then simplifying to show that the equation still holds true.
Proof by induction is a valid method for solving equations because it is based on the principle of mathematical induction, which states that if a statement holds true for a starting value and the inductive step can be proven, then the statement must be true for all natural numbers. This method is widely accepted and used in mathematics due to its logical and systematic approach.