Prove that 2n+10 +n is O(2n)

Expert Solution

because 2n+10 +n is O(2n) doesn't makes sense

1)
f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0

2^(n+10) + n = O(2^n)
=>  2^(n+10) + n <= c(2^n)
Let's assume c = 1025
=>  2^(n+10) + n <= c(2^n)
=>  2^(n+10) + n <= 1025(2^n)
=>  2^10*2^n + n <= 1025(2^n)
=>  1024(2^n) + n <= 1025(2^n)
=>  n <= (2^n)
it's true for all n >= 1

so, 2^(n+10) + n = O(2^n) given c = 1025 and n0 = 1

2)
In case if it is just 2n+10 +n is O(2n), use this
f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0

2n+10 +n is O(2n)
=>  3n+10 is O(2n)
=>  3n+10 <= c(2n)
Let's assume c = 2
=>  3n+10 <= c(2n)
=>  3n+10 <= 2(2n)
=>  3n+10 <= 4n
=>  10 <= n
it's true for all n >= 10

so, 2n+10 +n is O(2n) given c = 2 and n0 = 10

venereology answered 13 hours ago

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