Prove that, for integer values of n such that 0 ≤ n < 4 $$2^{n+2} > 3^n$$ - AQA - A-Level Maths Mechanics - Question 5 - 2020 - Paper 1

For n = 1:

$2^{1+2} = 2^3 = 8$ $3^1 = 3$

Thus, $8 > 3$ is true.

For n = 2:

$2^{2+2} = 2^4 = 16$ $3^2 = 9$

Thus, $16 > 9$ is true.

For n = 3:

$2^{3+2} = 2^5 = 32$ $3^3 = 27$

Thus, $32 > 27$ is true.

Since we have verified that for all integer values of n such that 0 ≤ n < 4:

• For n = 0: $2^{2} > 3^0$
• For n = 1: $2^{3} > 3^1$
• For n = 2: $2^{4} > 3^2$
• For n = 3: $2^{5} > 3^3$

This proves that $2^{n+2} > 3^n$ holds true for all specified integer values of n.