B1. The real solutions of the equation $x^{4}-x^{3}-2 x-4=0$ are the first two terms of a decreasing arithmetic sequence with 40 terms.

a) Calculate the first two terms and the difference of the sequence.

b) Calculate the last term and write the general term $a_{n}$ of the sequence.

c) Calculate the sum of all terms of the sequence with odd indices.
B1.

a) We find the real solutions of the given equation and write down the first two terms of the decreasing arithmetic sequence $a_{1}=2$ and $a_{2}=-1$. The difference of the sequence is $d=-3$.

b) We calculate the 40th term of the sequence $a_{40}=a_{1}+39 d=2+39 \cdot(-3)=-115$. We write the general term of the sequence $a_{n}=a_{1}+(n-1) d=2+(n-1) \cdot(-3)=5-3 n$.

c) We determine that the difference between the terms of the sequence with odd indices is $d=-6$ and that the given series has 20 terms. We use the formula for the sum of 20 terms of an arithmetic sequence and get $s_{20}=$ $\frac{20 \cdot(2 \cdot 2+19 \cdot(-6))}{2}=-1100$.
Writing $a_{1}=2, a_{2}=-1$ ..... 2 points
Writing $d=-3$ ..... 1 point
Calculating $a_{40}=-115$ ..... 1 point
Writing $a_{n}=5-3 n$ ..... 1 point
Writing or considering $d=-6$ ..... 1 point
Writing or considering that we are summing 20 terms ..... 1 point
Calculating $s_{20}=-1100$ ..... 1 point