SOLUTION: Prove the following using mathematical induction. Be sure to show the anchor as well as the full induction steps. Please show all the parts to the proof. 3+7+11+...+(4n-1)=n(2n+

Algebra ->  Sequences-and-series -> SOLUTION: Prove the following using mathematical induction. Be sure to show the anchor as well as the full induction steps. Please show all the parts to the proof. 3+7+11+...+(4n-1)=n(2n+      Log On


   



Question 1137605: Prove the following using mathematical induction. Be sure to show the anchor as well as
the full induction steps. Please show all the parts to the
proof.
3+7+11+...+(4n-1)=n(2n+1)

Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Base case: n=1
LHS: 3
RHS: 1(2*1+1) = 1(3) = 3


Hypothesis: 3+7+11+...+(4k-1) = k(2k+1)


Must show the above is true for n=k+1
3+7+11+...+(4k-1)+(4(k+1)-1) =
+green%283%2B7%2B11%29+ +green%28matrix%281%2C3%2C%22%2B%22%2C%22...%22%2C%22%2B%22%29%29 +green%28%284k-1%29%29+ + (4(k+1)-1)
By hypothesis, the green terms are k(2k+1), so we write
= +k%282k%2B1%29+%2B+%284%28k%2B1%29-1%29+
= %282k%5E2+%2B+k%29+%2B+%284k%2B3%29+
= +2k%5E2+%2B+5k+%2B+3+
= +%28k%2B1%29%282k%2B3%29+
(n=k+1)
= +++n%282n%2B1%29+ DONE