2. Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence defined by: $a_{1}=\frac{1}{2}$ and $a_{n+1}+a_{n}=\frac{2}{n^{2}+2 n}, \forall n \geq 1$.

a. Determine the general term of the sequence.

b. Compute the sum $S=\sum_{k=1}^{n}(2 k+1) a_{k}^{2}$ and show that $S<1$.

Prof. Traian Tămâian
2. a) $a_{2}=\frac{1}{2 \cdot 3}$ and $a_{8}=\frac{1}{3 \cdot 4}$

$\mathbf{1 p}$

$P(n): \quad a_{n}=\frac{1}{n \cdot(n+1)}$

## $1 p$

Proof by induction

$2 p$

$$
=\sum_{1}^{n} \frac{2 k+1}{k^{2}(k+1)^{2}}=1-\frac{1}{(n+1)^{2}} \times 1
$$