Use mathematical induction to prove that the statement holds for all positive integers. Also, label the basis, hypothesis, and induction step.
1 + 5 + 9 + … + (4n -3)= n(2n-1)
for k=1: 1 = 1(2-1)
assume for k
for n=k+1,
1+5+...+(4k-3)+(4(k+1)-3) = k(2k-1) + (4(k+1)-3)
= k(2k-1) + (4k+1)
= 2k^2 - k + 4k + 1
= 2k^2 + 3k + 1
= (k+1)(2k+1)
= (k+1)(2(k+1)-1)
To prove the given statement using mathematical induction, we need to follow three steps: the basis step, the induction hypothesis, and the induction step.
1. Basis Step:
We start by verifying that the statement holds true for the first positive integer. In this case, we will check if the statement holds for n = 1.
Plugging in n = 1, we have:
1 + 5 + 9 + … + (4(1) - 3) = 1(2(1) - 1)
1 + 5 + 9 - 3 = 1(2-1)
12 = 1
Since 12 is not equal to 1, the statement does not hold for the basis step.
Therefore, we cannot proceed with mathematical induction for this statement.
If you have any other questions or need further assistance, feel free to ask.