The ratio of 7th and 5th terms of expansion (square root 2 +1)^n is 2. What is n?

We'll write the formula of the general term of the
expansion:

Tk+1 =
C(n,k)*a^(n-k)*b^k

The 7th term
is:

T7 =
C(n,6)*(sqrt2)^(n-6)*1^6

The 5th term
is:

T5 =
C(n,4)*(sqrt2)^(n-4)*1^4

T7/T5 =
2

C(n,6)*(sqrt2)^(n-6)/C(n,4)*(sqrt2)^(n-4) =
2

C(n,6) = n!/6!(n-6)!

C(n,4)
= n!/4!(n-4)!

C(n,6)/C(n,4) =
4!(n-4)!/6!(n-6)!

C(n,6)/C(n,4) =
4!(n-6)!(n-5)(n-4)/4!*5*6(n-6)!

We'll simplify and we'll
get:

C(n,6)/C(n,4) =
(n-5)(n-4)/5*6

C(n,6)/C(n,4) =
(n-5)(n-4)/30

(sqrt2)^(n-6-n+4) = (sqrt2)^-2 =
1/2

(n-5)(n-4)/30*2 =
2

(n-5)(n-4) = 120

We'll
remove the brackets:

n^2 - 9n + 20 - 120 =
0

n^2 - 9n - 100 = 0

We'll
apply quadratic formula:

n1 = [9+sqrt(81 +
400)]/2

Since n is not a natural number, the
equation has no solution.