If e^ x +e^ 5 x e^ 3 x =a₀+a₁ x+a₂ x²+a₃ x³+ then the value of 2 a₁+2³ a₃+2⁵ a₅+ is

Step-by-step solution

Let e ^ x + e ^ 5 x e ^ 3 x = a ₀+ a ₁ x + a ₂ x ²+ a ₃ a ³+ . aligned &= e ^ x e ^ 3 x + e ^ 5 x e ^ 3 x = a ₀+ a ₁ x + a ₂ x ²+ . =& e ^ -2 x + e ^ 2 x = a ₀+ a ₁ x + a ₂ x ²+ a ₃ x ³+ . aligned By using e ^ x =1+ x + x ² 2 ! + x ³ 3 ! + _ -- -- and e ^ - x =1- x + x ² 2 ! - x ³ 3 ! +--- e ^ -2 x + e ^ 2 x =2 [1+ (2 x )² 21 + (2 x )⁴ 41 + . ] array l =a₀+a₁ x+a₂ x²+a₃ a³+ . =a₁=a₃=a₅= =0 array Hence, 2 a ₁+2³ a ₃+2⁵ a ₅+ .=0