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I have recently been introduced to mathematical induction in my discrete mathematics course. This is the first time I’ve ever seen induction, and so of course I was very confused. I am still confused, but not as much as before. I’m doing problems out of the book for practice, but I’m not sure if this is correct so far. I’m also not sure how to proceed with the inductive step. If someone could critique my work and guide me through the rest of the problem that would be great. On a side note, do you guys think that I might be in over my head with a discrete mathematics course? I’m fresh out of high school, and the last math class I took was pre-calculus. I wanted to get a head start on college, but now that I’m reaching induction, recursion, etc… I feel like I cannot keep up with the class. A lot of the other people in my class seem to be way more advanced as far as understand computer science and math, and so I begin to wonder whether or not I made a mistake. Thanks guys.
Let P(n) be the statement that 1^2 + 2^2 + … + n^2 = n(n+1)(2n+1)/6
for the positive integer n.
a)What is the statement P(1)?
P(1) = 1(1+1) (2(1)+1)/6 = 1(2)(3)/6 = 6/6 = 1
b)Show that P(1) is true, completing the basis step of the proof
1^2 = 1, and since the above equation results in 1. Therefore, P(1) is true. c)What is the inductive hypothesis?
(A) : “1^2 + 2^2 + … + k^2 = k(k+1)(2k+1)/6” where k is any positive integer.
(B): “1^2 + 2^2 +…+ k^2 + (k+1)^2 = (k+1)(k+1) (2 (k+1) +1)/6”
Assuming A, we will prove B. d)What do you need to prove in the inductive step?
Assuming that A is true, you need to prove B. e) Complete the inductive step
Let P(n) be the statement that 1^2 + 2^2 + … + n^2 = n(n+1)(2n+1)/6
for the positive integer n.
a)What is the statement P(1)?
P(1) = 1(1+1) (2(1)+1)/6 = 1(2)(3)/6 = 6/6 = 1
b)Show that P(1) is true, completing the basis step of the proof
1^2 = 1, and since the above equation results in 1. Therefore, P(1) is true. c)What is the inductive hypothesis?
(A) : “1^2 + 2^2 + … + k^2 = k(k+1)(2k+1)/6” where k is any positive integer.
(B): “1^2 + 2^2 +…+ k^2 + (k+1)^2 = (k+1)(k+1) (2 (k+1) +1)/6”
Assuming A, we will prove B. d)What do you need to prove in the inductive step?
Assuming that A is true, you need to prove B. e) Complete the inductive step
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