To prove that 7 n - 3 n is divisible by 4 for all natural numbers n, the inductive step writes 7 k+1 - 3 k+1 as:

7 x 7 k - 3 x 3 k only, with no simplification

To prove that 7 n - 3 n is divisible by 4 for all natural…

Answer: 7(7^k - 3^k) + 4(3^k).

A 7(7^k - 3^k) + 4(3^k)
B 7^k - 3^k + 4
C 7 x 7^k - 3 x 3^k only, with no simplification
D 4(7^k - 3^k)

Correct answer: A. 7(7^k - 3^k) + 4(3^k)

Explanation: 7^k+1 - 3^k+1 = 7 x 7^k - 3 x 3^k = 7(7^k - 3^k) + 7(3^k) - 3(3^k) = 7(7^k-3^k) + 4(3^k). Since 7^k-3^k is divisible by 4 (hypothesis) and 4(3^k) is clearly divisible by 4, the whole expression is divisible by 4.

Mathematical induction works like a row of dominoes: proving the base case P(1) tips the first domino, and proving the inductive step (P(k) ⟹ P(k+1)) guarantees each domino knocks over the next - together these two facts guarantee ALL dominoes fall, without checking each one individually.

Concept context

A proof technique used to establish that a statement is true for every natural number, using a base case and an inductive step.

Read the full Principle of Mathematical Induction notes →

To prove that 7n - 3n is divisible by 4 for all natural numbers n, the inductive step writes 7k+1 - 3k+1 as:

Concept context

More Principle of Mathematical Induction questions

To prove that 7ⁿ - 3ⁿ is divisible by 4 for all natural numbers n, the inductive step writes 7^k+1 - 3^k+1 as: