Wednesday, April 21, 2010

`1 + 4 + 7 + 10 + ... (3n - 2) = n/2 (3n - 1)` Use mathematical induction to prove the formula for every positive integer n.

You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that:


Step 1: Basis: Show that the statement P(n) hold for n = 1, such that:


`1 = 1/2*(3*1-1) => 1 =2/2 => 1=1`


Step 2: Inductive step: Show that if P(k) holds, then also P(k + 1) holds:


`P(k): 1 + 4 + 7 + .. + (3k-2) = (k(3k-1))/2 ` holds


`P(k+1):  1 + 4 + 7 + .. + (3k-2) + (3k+1) =  ((k+1)(3k+2))/2`


You need to use induction hypothesis that P(k) holds, hence, you need to re-write the left side, such that:


`(k(3k-1))/2 + (3k+1) = ((k+1)(3k+2))/2`


`3k^2 - k + 6k + 2 = 3k^2 + 2k + 3k + 2`


You need to add the like terms, such that:


`3k^2 + 5k + 2 = 3k^2 + 5k + 2`


Notice that P(k+1) holds.


Hence, since both the basis and the inductive step have been verified, by mathematical induction, the statement `P(n): 1 + 4 + 7 + .. + (3n-2) = (n(3n-1))/2`  holds for all positive integers n.

No comments:

Post a Comment

How does the choice of details set the tone of the sermon?

Edwards is remembered for his choice of details, particularly in this classic sermon. His goal was not to tell people about his beliefs; he ...