Prove the statement by using the Principle of Mathematical Induction: 1 + 2 + 22 + ... + 2n = 2n+1 – 1 for all natural numbers n.

प्रश्न

Prove the statement by using the Principle of Mathematical Induction:

1 + 2 + 2² + ... + 2ⁿ = 2ⁿ⁺¹ – 1 for all natural numbers n.

प्रमेय

उत्तर

P(n) is 1 + 2 + 2² + … 2ⁿ = 2ⁿ⁺¹ – 1.

So, substituting different values for n, we get,

P(0) = 1 = 2⁰⁺¹ − 1 Which is true.

P(1) = 1 + 2 = 3 = 2¹⁺¹ − 1 Which is true.

P(2) = 1 + 2 + 2² = 7 = 2²⁺¹ − 1 Which is true.

P(3) = 1 + 2 + 2² + 2³ = 15 = 2³⁺¹ − 1 Which is true.

Let P(k) = 1 + 2 + 2² + … 2^k = 2^k+1 – 1 be true;

So, we get,

⇒ P(k + 1) is 1 + 2 + 2² + … 2^k + 2^k+1 = 2^k+1 – 1 + 2^k+1

= 2 × 2^{k + 1} – 1

= `2^(("k" + 1) + 1` – 1

⇒ P(k + 1) is true when P(k) is true.

Therefore, by Mathematical Induction, 1 + 2 + 2² + … 2ⁿ = 2ⁿ⁺¹ – 1 is true for all natural numbers n.

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Principle of Mathematical Induction

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Q 15 Q 14 Q 16

अध्याय 4: Principle of Mathematical Induction - Exercise [पृष्ठ ७१]

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एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 11

अध्याय 4 Principle of Mathematical Induction
Exercise | Q 15 | पृष्ठ ७१

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Principle of Mathematical Induction

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