Maths · Binomial theorem for a positive integral index
Assertion ^{n} \) can be expressed as for and Reason ^{n} \) can be written in t
Assertion \( (\sqrt{2}-1)^{n} \) can be expressed as \( \sqrt{N} \) \( \sqrt{N-1} \) for \( \forall N>1 \) and \( n \in N \) Reason \( (\sqrt{2}-1)^{n} \) can be written in the form \( \boldsymbol{\alpha}+\boldsymbol{\beta} \sqrt{\boldsymbol{2}} \forall, \boldsymbol{\alpha}, \boldsymbol{\beta} \) are integers \& n is a positive integer.
- A. Both Assertion \& Reason are individually true \& Reason is correct explanation of Assertion,
- B. Both Assertion \& Reason are individually true but Reason is not the correct (proper) explanation of Assertion,
- C. Assertion is true but Reason is false
- D. Assertion is false but Reason is true.
Step-by-step solution
Assertion: (√2-1)^n can be written as √N - √(N-1) for some N>1. This is true because squaring gives (2N-1) - 2√(N(N-1)) which equals (a+b√2)^2 with integers a,b, and such representation exists for all n. Reason: (√2-1)^n = α+β√2 with integers α,β by binomial theorem. Both are true, but Reason does not directly lead to the √N-√(N-1) form; extra steps are required. Hence B.
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