Use mathematical induction to prove the statement is true for all positive integers n.n ? 2n

What will be an ideal response?

S_n:

n ? 2ⁿ

Step 1: Determine if S₁, S₂, and S₃ are true.

Step 2:

Assume S_k to be true, where k is a natural number, then

S_k: k ? 2^k

Step 3: Find S_k+1.
S_k+1: k + 1 ? 2^k+1

Step 4: Truth of S_k implies that S_k+1 is true.
   k ? 2^k  This is S_k
   2k ? 2*2^k  Mult both sides by 2
   2k ? 2^k+1  Adding exponents of right
   k + k ? 2^k+1   Substituting k + k for 2k
Since k is any natural number, 1 ? k,
then k + 1 ? k + k by adding k to both sides.
k + 1 ? 2^k+1   Substituting k + 1 for k + k.
  This is S_k+1.

Mathematics