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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