Let , , and be the coefficients of x^7, x^5, x^3 and x respectively in the expansion of (x+ x^3-1 )^5+ (x- x^3-1 )^5, x 1 . If u and v satisfy the equations aligned & u+ v=18 & u+ v=20 aligned then u+v equals :

Step-by-step solution

aligned & (x+ x^3-1 )^5+ (x- x^3-1 )^5 & - [ ^5 C₀ x^5+ ^5 C₁ x^4 ( x^3-1 )+ + ^5 C₅ ( x^3-1 )^5 ]+ & [ ^5 C₀ x^5- ^5 C₁ x^4 ( x^3-1 )+ .+ ⁻⁵ C₅ ( x^3-1 )^5 ] & =2 [x^5+ ^5 C₂ x^3 (x^3-1 )+ ^5 C₄ (x^3-1 )^2 ] & =10 x^7+20 x^6+2 x^5-20 x^4-20 x^3+10 x aligned Now =10, =2, =-20, =10 Also, aligned & . array c 10 u+2 v=18 -20 u+10 v=20 array u=1, v=4 & u+v=5 aligned