The sum of the first 5 natural numbers is 15 or (5)(6)/2. The sum of the first 21 natural numbers is 231=(21)(22)/2, and the sum of the first 30 numbers is 465. Use inductive reasoning to derive the formula for the sum of the first n natural numbers.
sum of first 5 natural numbers = (5)(6)/2
sum of first 21 natural numbers = (21)(22)/2
sum of first 30 natural numbers = 465 = (30)(31)/2
sum of first n natural numbers = (n)(n+1)/2 -- provable by induction
To derive the formula for the sum of the first n natural numbers, we can use the method of mathematical induction. Mathematical induction is a principle in mathematics used to prove statements for all positive integers n.
Step 1: Base case
Let's start by verifying the formula for a small value of n, such as 1.
Since the sum of the first 1 natural number is 1 itself, we have: 1 = (1)(1+1)/2, which is true.
Step 2: Inductive hypothesis
Assume that the formula holds true for some positive integer k. In other words, assume that the sum of the first k natural numbers is given by k(k+1)/2.
Step 3: Inductive step
Now, we need to prove that the formula also holds true for k+1. In other words, we need to show that the sum of the first (k+1) natural numbers can be expressed as (k+1)(k+1+1)/2.
Let's assume the sum of the first k natural numbers is S(k) = k(k+1)/2.
To find the sum of the first (k+1) natural numbers, we need to add the (k+1)st natural number to S(k).
So, the sum of the first (k+1) natural numbers is S(k+1) = S(k) + (k+1).
By substituting the inductive hypothesis, we get:
S(k+1) = k(k+1)/2 + (k+1).
Now, we can simplify this expression:
S(k+1) = k(k+1)/2 + (2(k+1))/2
Expanding and collecting like terms, we have:
S(k+1) = [k(k+1)+2(k+1)]/2
Factoring out (k+1), we get:
S(k+1) = (k+1)(k+2)/2
This is the same form as the original formula, but with k replaced by (k+1). Therefore, the formula holds true for (k+1) as well.
Step 4: Conclusion
We have shown that if the formula holds true for some positive integer k, then it also holds true for (k+1). Since we verified the formula for the base case n=1, we can conclude that the formula holds true for all positive integers n.
Hence, the sum of the first n natural numbers can be expressed as: S(n) = n(n+1)/2.