The Pivotals of Mathematics

Algebra Level 5

$$a+b+c=\lambda$$ for reals $$a,b,c$$ . We define $$\displaystyle t_n= \sum \limits_{cyc} a^{2n+1}$$ for $$n\in \text{N}$$. Further it's given that $$t_1={\lambda}^3$$. Find the value of $$\lambda$$ if $$\displaystyle \sum_{n=1}^{\infty} t_n=\frac{1}{6}$$.

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