can someone please show me step by step how to do 4 Prove th

can someone please show me step by step how to do 4?

Prove that 2n 2 ns for all integers n 10. Is 10 the least positive inte no such that 2\" na for all integers n 2 no?

Solution

Let us consider a number less than 10 say 9

2^9 = 512

9^3 = 729 >2^9

For n =8 also 2^8 = 256 and 8^3 = 512 >2^8

Thus for n = 1,2,...9, 2^n <n^3

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n =10

2^10 = 1024

10^3 = 1000

Thus 2^10>10^3

Let us assume that this holds good for k >10

To prove that holds good for k+1

2^k >k^3

Hence 2^k+1 >2k³

(k+1)³ = k^3 +1+3k(k+1)

So 2(k+1)^{3 =} 2k³+2+3k(k+1)

< 2(2^k) + 3k^2+3k+2

<2(2^k)

Thus proved by induction for n >10