Given:
x+(1/x)= square root of 3
PROVE:
(x^13)+(1/(x^13))= square root of 3
To prove the given statement, we can use mathematical induction.
Step 1: Verify the statement for the base case (n = 1)
Substitute n = 1 into the given equation:
x + (1/x) = √3
Step 2: Assume the statement is true for n = k, where k is any positive integer.
Assume that (x^k) + (1/(x^k)) = √3
Step 3: Prove the statement is true for n = k + 1
We need to prove that (x^(k+1)) + (1/(x^(k+1))) = √3
Starting with the statement we assumed in Step 2:
(x^k) + (1/(x^k)) = √3
Multiplying both sides by (x + (1/x)), we get:
(x + (1/x)) * ((x^k) + (1/(x^k))) = √3 * (x + (1/x))
Expanding the left-hand side of the equation:
(x * (x^k)) + (x * (1/(x^k))) + ((1/x) * (x^k)) + ((1/x) * (1/(x^k))) = √3 * (x + (1/x))
Simplifying the equation:
(x^(k+1)) + (1/(x^(k-1))) + (x^(k-1)) + (1/(x^(k+1))) = √3 * (x + (1/x))
Rearranging the terms:
(x^(k+1)) + (1/(x^(k+1))) + (x^(k-1)) + (1/(x^(k-1))) = √3 * (x + (1/x))
Now, let's substitute the given equation (x + (1/x) = √3) into the right-hand side of the equation, we have:
(x^(k+1)) + (1/(x^(k+1))) + (x^(k-1)) + (1/(x^(k-1))) = √3 * √3
Simplifying further:
(x^(k+1)) + (1/(x^(k+1))) + (x^(k-1)) + (1/(x^(k-1))) = 3
Finally, we can observe that the equation we ended up with is equivalent to the statement we assumed for n = k + 1. Hence, we have proved that if the statement is true for n = k, then it is also true for n = k + 1.
By using mathematical induction, we have proven that for any positive integer n, (x^n) + (1/(x^n)) = √3.