📐 Mathematics · Principle of Mathematical Induction · JEE

To prove that 10n + 3 x 4n+2 + 5 is divisible by 9 for all natural numbers n, what is the value at n=1, and is it divisible by 9?

Answer: 10 + 192 + 5 = 207, and 207/9 = 23, so yes.

A 10 + 192 + 5 = 207, and 207/9 = 23, so yes
B 10 + 48 + 5 = 63, and 63/9 = 7, so yes
C 207, but it is not divisible by 9
D 10 + 3 + 5 = 18, divisible by 9

Correct answer: A. 10 + 192 + 5 = 207, and 207/9 = 23, so yes

Explanation: At n=1: 101 + 3 x 43 + 5 = 10 + 3(64) + 5 = 10 + 192 + 5 = 207. Dividing, 207/9 = 23 exactly, confirming the base case.

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.

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To prove that 10<sup>n</sup> + 3 x 4<sup>n+2</sup> + 5 is divisible by 9 for all natural numbers n, what is the value at n=1, and is it divisible by 9?

Concept context

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