Prove 3+4+5+...+(n+2) = [n(n+5)]/2 for n>4
Do the first step in a proof by induction.
You must have a typo
in 3+4+5+ ... + (n+2)
the general term does not match your first three numbers
I'm not sure. It was on my math test. I typed it out exactly like it said on the test.
To prove the given statement using mathematical induction, we need to follow these steps:
Step 1: Base Case
Show that the statement holds true for the smallest possible value of n.
In our case, the smallest value of n is 5 (since n > 4). So, let's substitute n = 5 into the given equation and simplify:
3 + 4 + 5 + ... + (5 + 2) = [|(|5|)|(|5 + 5|)]/2
= [5(10)]/2
= 25
Therefore, for n = 5, the equation holds true.
Step 2: Inductive Hypothesis
Assume that the statement holds true for some arbitrary positive integer k. That is, assume 3 + 4 + 5 + ... + (k + 2) = [k(k + 5)]/2.
Step 3: Inductive Step
Now, we need to prove that if the statement holds true for k, then it also holds true for k + 1.
Let's consider the left side of the equation for k + 1: 3 + 4 + 5 + ... + [(k + 1) + 2].
By assuming that the statement holds true for k, we can rewrite the above expression as:
[k(k + 5)]/2 + (k + 1 + 2)
We simplify this expression:
[k(k + 5)]/2 + (k + 3)
= [k(k + 5) + 2(k + 3)]/2
= [k² + 5k + 2k + 6]/2
= [(k² + 7k + 6)]/2
= [(k + 1)(k + 6)]/2
Thus, for k + 1, the equation holds true.
Therefore, by applying mathematical induction, we have proven that 3 + 4 + 5 + ... + (n + 2) = [n(n + 5)]/2 for n > 4.