- #1
carameled
- 3
- 0
Prove by math induction that
n
sigma 3i + 1 = n/2 (3n + 5)
i = n
n
sigma 3i + 1 = n/2 (3n + 5)
i = n
MarkFL said:I think what you mean is the induction hypothesis \(P_n\):
\(\displaystyle \sum_{i=1}^{n}\left(3i+1\right)=\frac{n}{2}(3n+5)\)
The first thing we want to do is confirm the base case \(P_1\) is true:
\(\displaystyle \sum_{i=1}^{1}\left(3i+1\right)=\frac{1}{2}(3(1)+5)\)
Is this true?
carameled said:Wow, well I'm just asking for the prove with math induction. I don't understand any of that..
MarkFL said:You don't understand what an induction hypothesis is, or demonstrating the truth of the base case? These are fundamental to induction. What method have you been taught?
Sigma notation is a shorthand way of representing a sum of terms in a sequence. It is commonly used in mathematical induction to express the general form of a summation, making the process more efficient and easier to understand.
In a proof by mathematical induction, the first step is to express the general form of the summation using sigma notation. Then, the base case is typically shown to be true, followed by the inductive step where the formula is applied to the next term. Finally, the proof is concluded by showing that the formula holds for all natural numbers.
Sure, let's say we want to prove that the sum of the first n even numbers is equal to n(n+1). The first step would be to write the general form using sigma notation: ∑_{k=1}^{n} 2k. Then, we show that this formula holds for the base case n=1, where the sum is 2*1=2, which is indeed equal to 1(1+1). Next, we assume that the formula holds for some arbitrary integer k, and then show that it also holds for the next integer k+1. Finally, the proof is concluded by showing that the formula holds for all natural numbers.
One common mistake is to assume that the formula holds for all natural numbers without properly showing the inductive step. It is important to show that the formula holds for the next integer after the assumed case, and not just for the assumed case itself. Another mistake is to incorrectly write the general form using sigma notation. It is important to carefully consider the starting and ending values of the summation, as well as the expression being summed.
No, mathematical induction with sigma notation is a powerful tool that can be used for a wide range of mathematical proofs beyond just summation formulas. It can also be used to prove statements about sequences, series, and other mathematical concepts. It is a valuable technique for proving many different types of mathematical statements.