Let `a=Sigma_(n=0)^(oo) (x^(3n))/((3n))!, b =Sigma_(n=1)^(oo) (x^(3n-2))/(3n-2)!` and
`C=Sigma_(n=1)^(oo) (x^(3n-1))/(3n-1)!` and w be a complex cube root of unity
Statement 1: a+b+c`=e^(x),a+bw+cw^(2)=e^(wx) and a+bw^(2)+cw=e^(w^(2))`
Statement 2: `a^(3)+b^(3)+C^(3)-3abc=1`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

We have
`a+b+c=underst(n=0)overset(infty)Sigma (x^(3n))/((3n))!+underset(n=1)overset(infty)Sigma (x^(3n-2))/(3x-2)!+underset(n=1)overset(infty)Sigma(x^(3n-1))/(3n-1)!`
`rarr a+b+c=1+x+(x^(2))/(2!)+(x^(3))/(3!)+(x^(4))/(4!)+(x^(5))/(5!)+..`
`rarra+b+c=e^(x)`
`rarr a+bw+cw^(2)=1+wx+(wx)^(2)/(2!)+(wx)^(3)/(3!)+..`
`rarr a+bw+cw^(2)=e^(wx)`
and `a+bw^(2)+cw=e^(w^(2))x`
So statemnet 1 is true
Now
`a^(3)+b^(3)+c^(3)-3abc=(a+b+c)(a+bw+cw^(2))(a+bw^(2)+cw)`
`rarr a^(3)+b^(3)+c^(3)-3abc=(a+b+c)(a+bw+cw^(2)))(a+bw^(2)+cw)`
`rarr a^(3)+b^(3)+c^(3)-3abc=e^(x)xxe^(wx)xxe^(w^(2)x)=e(1+w+w^(2))x=e^(0x)=1`
So statement 2 true
Now that statemnt 2 is not a correct explanation for statement1