If a+1/a = 4, what is the of a4+1/a4 ?

Question

IF a + 1/a = 4 . Then what is the value of a⁴ + 1/a⁴ ?

Answer 1

First, notice that (a + 1/a)² = 4² = 16.  But also, (a + 1/a)² = a² + 2a(1/a) + 1/a² = a² + 2 + 1/a² = a² +  1/a² + 2

 

So a² + 1/a² + 2 = 16, so a² + 1/a² = 14.

 

Now, do that again...

 

(a² + 1/a²)² = 14² = 196, but also (a² + 1/a²)² = a⁴ + 2(a²)(1/a²) + 1/a⁴ = a⁴ + 2 + 1/a⁴ = a⁴ + 1/a⁴ + 2

 

So a⁴ + 1/a⁴ + 2 = 196, so a⁴ + 1/a⁴ = 194

 

You can see the pattern...  E.g., a⁸ + 1/a⁸ = 194² - 2, etc.