Final answer:
To prove the given statement using mathematical induction, we need to show that it holds true for the base case and then prove it for the next case using the induction hypothesis. By following the steps of mathematical induction, we can validate the given equation for all positive integers n.
Step-by-step explanation:
To prove the statement using mathematical induction, we need to follow these steps:
Show that the statement is true for the base case, which is usually n = 1.
Assume that the statement is true for a particular value of n, which is called the induction hypothesis.
Using the induction hypothesis, prove that the statement is true for n+1.
Step 1:
When n = 1, the left side of the equation is 10, and the right side is 5(1)(1+1) = 15. So, the statement is true for n = 1.
Step 2:
Assume that the statement is true for some k, where k is a positive integer. That is, assume that 10+20+30+...+10k = 5k(k+1).
Step 3:
Using the induction hypothesis, we need to prove that the statement is true for k+1. That is, we need to prove that 10+20+30+...+10k+10(k+1) = 5(k+1)(k+1+1).
Starting with the left side of the equation:
10+20+30+...+10k+10(k+1) = 10+20+30+...+10k+10k+10
Using the induction hypothesis, we can substitute 5k(k+1) for the sum of the first k terms:
10+20+30+...+10k+10k+10 = 5k(k+1)+10k+10
Simplifying the equation:
5k(k+1)+10k+10 = 5(k+1)(k+1+1)
Expanding:
5k(k+1)+10k+10 = 5(k+1)(k+2)
5k²+5k+10k+10 = 5(k+1)(k+2)
Combining like terms:
5k²+15k+10 = 5(k+1)(k+2)
Factoring out a 5:
5(k²+3k+2) = 5(k+1)(k+2)
Dividing both sides by 5:
k²+3k+2 = (k+1)(k+2)
Simplifying the right side of the equation:
k²+3k+2 = k²+3k+2
Since the left side equals the right side, the statement is true for k+1.
Therefore, the statement is true for all positive integers n.