If n is an odd positive integer and (1+x+x²+x³ )^ n = _ r=0 ^ 3 n a_ r x^ r , then a₀-a₁+a₂-a₃+ -a_ 3 n is equal to

Step-by-step solution

Given, (1+x+x²+x³ )^ n = _ r=0 ^ 3 n a_ r x^ r and n is an odd positive integer. aligned & [(1+x) (1+x² ) ]^ n = _ r=0 ^ 3 n a_ r x^ r & (1+x)^ n (1+x² )^ n = _ r=0 ^ 3 n a_ r x^ r aligned If we take n=1 , gathered (1+x+x²+x³ )= _ r=0 ³ a_ r x^ r =a₀+a₁ x+a₂ x²+a₃ x³ gathered On comparing both sides, a₀=1, a₁=1, a₂=1, a₃=1 (i) If we take n=3 , gathered (1+x)³ (1+x² )³= _ r=0 ⁹ a_ r x^ r (1+x³+3 x²+3 x ) (1+x⁶+3 x⁴+3 x² ) = _ r=0 ⁹ a_ r x^ r (1+x³+3 x²+3 x+x⁶+x⁹ . +3 x⁸+3 x⁷+3 x⁴+3 x⁷+9 x⁶ . +9 x⁵+3 x²+3 x⁵+9 x⁴+9 x