JEE Main2012MathematicsBinomial TheoremActual
If n is a positive integer, then ( 3 +1)^ 2 n -( 3 -1)^ 2 n is
Options
- Aan irrational number
- Ban odd positive integer
- Can even positive integer
- Da rational number other than positive integers
Correct answer
A. an irrational number
Step-by-step solution
aligned & ( 3 +1)^ 2 n -( 3 -1)^ 2 n = [( 3 +1)^2 ]^n- [( 3 -1)^2 ]^n=(4+2 3 )^n-(4-2 3 )^n & =2^n [(2+ 3 )^n-(2- 3 )^n ] & =2^n [ ^n C₀ 2^n+ ^n C₁ 2^ n-1 3 + ^n C₂ 2^ n-2 3+ ]- [ ^n C₀ 2^n- ^n C₁ 2^ n-1 3 + ^n C₂ 2^ n-2 3- ] & =2^ n+1 [ ^n C₁ 2^ n-1 3 + ^n C₃ 2^ n-3 3 3 + ]=2^ n+1 3 (some integer) aligned Which is irrational