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Maths Binomial Theorem & Mathematical Induction Mix Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 12, 2026

If U n = ( + 1) 2n + ( – 1) 2n then-

4U n+1 = 2U n – 3U n–1

U n+1 = 4U n – U n–1

U n+1 = 8U n – 4U n–1

U n+1 = 2U n

Text Solution

Verified by Experts

The correct answer is:

U n = [( + 1) 2 ] n + [( – 1) 2 ] n

= (4 + 2 ) n + (4 – 2 ) n

= α n + β n where α + β = 8, αβ = 4

8U n = ( α + β ) ( α n + β n )

= α n+1 + β n+1 + αβ n + βα n

= U n+1 + αβ ( α n–1 + β n–1 )

= U n+1 + 4U n–1

⇒ U n+1 = 8U n – 4U n–1.

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