+10 I'm trying to solve N from \(\frac{n * (n + 1)}{2}\)
Let's say N = 4, thus the result is (1 + 2 + 3 + 4) ---> 10.
Steps:
\(\frac{n * (n + 1)}{2} = 10\)
Multiply both sides by 2
\(n * (n + 1) = 20\)
n * (n + 1) becomes = \({2n}^{2}\)
so \({2n}^{2} = 20\)
divide both sides by 2
\({n}^{2} = 10\)
take the square root
\(n = \sqrt{10} = 3.1622776601683793\)
Thus the result is wrong. Where's the mistake, and what are the correct steps instead?
Thanks!
+9665 The wrong step is at the step n*(n+1) = 2n^2.
n * (n + 1) = n^2 + n instead.
\(n(n+1) = n^2 + n\)
\(n^2 + n = 20\\ n^2 + n - 20 = 0\\ (n+5)(n-4) = 0\\ n = -5(\text{rejected})\text{ or }n=4\)
Now that's correct.
+9665 And n(n+1)/2 = 1 + 2 + ...... + n can be proved by mathematical induction.
Let P(n) be n(n+1)/2 = 1 + 2 + ....... + n
For n = 1,
LHS = 1(1+1)/2 = 1
RHS = 1 = LHS
P(1) is true.
Assume that P(k) is true,
For n = k + 1
LHS
= (k+1)(k+2)/2
RHS
= 1 + 2 + ...... + k + (k + 1)
= (k + 1)(k)/2 + (k + 1)
= \((\dfrac{k+1}{2})(k + 2)\)
= (k+1)(k+2)/2
= LHS
Therefore when P(k) is true, P(k + 1) is true.
By the definition of mathematical induction, P(n) is true for all positive integers n.
