Let P = (5 + 8) 2n+1 and f = P –[P], where [.] denotes the…

Maths Binomial Theorem & Mathematical Induction Mix Subjective Type

Published on: August 13, 2026

Let P = (5 + 8) 2n+1 and f = P –[P], where [.] denotes the greatest integer function. Find Pf.

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Sol.  P = (5 + 8) 2n+1 = [P] + f …(1)

Where 0 ≤ f < 1

Now let (5 –8) 2n+1 = f ′ … (ii)

Where 0 < f ′ < 1

Subtracting (1) and (2), we get

⇒ [P] + f– f ′ = (5 + 8) 2n+1 – (5 –8) 2n+1 = even integer

⇒ Hence f – f ′ is also an integer.

i.e. f – f ′ = 0 {  –1 < f – f ′ < 1}

∴ f = f ′ …(iii)

Now L.H.S. = Pf = Pf ′

from (iii)

= (5 + 8)

2n+1 (5 –8) 2n+1 = (75 –64)

2n+1 = 11

2n+1 R.H.S.

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