Find the value of n if 3^2n+1 - 4(3^n+1)+9=0

Answer 1

To solve the equation 3^(2n+1) - 4(3^(n+1)) + 9 = 0, let's separate the terms involving 3:

(3^(2n) * 3^1) - 4(3^n * 3^1) + 9 = 0

By simplifying the exponents:

3^(2n+1) - 4(3^(n+1)) + 9 = 0

Now, let's factor out 3^(n+1):

3^(n+1)(3^n - 4) + 9 = 0

Now, we have a quadratic-like equation in terms of 3^n. Let's substitute u = 3^n:

u - 4u + 9 = 0

Simplifying:

-3u + 9 = 0
-3u = -9
u = 3

Now, let's substitute back in our expression for u:

3^n = 3

Taking the logarithm of both sides of the equation:

log_3(3^n) = log_3(3)

n = 1

Therefore, the value of n is 1.