## Task 4 - 261214

For each real number $b$, let $\left(a_{n}\right)$ be the sequence defined by

$$
a_{n}=\frac{3 n+b}{2 n-1} \quad(n=1,2, \ldots)
$$

Determine all integers $b$ for which the sequence defined by (1) has exactly three terms that satisfy the inequalities

$$
1.45<a_{n}<1.47
$$

For each such integer $b$ (if any exist), provide the three terms $a_{n}$ that satisfy (2).
I. For every real number $b$, a term $a_{n}$ of the sequence defined by (1) satisfies (2) if and only if both

$$
1.450$ as $n$ runs through, (3) is equivalent to $2.9 n - 1.45 - 14.5 - 10 b
$$

furthermore, (4) is equivalent to $3 n + b b > -\frac{42}{20}
\end{aligned}
$$

The only integer $b$ that satisfies this inequality is $b = -2$. Therefore, only this number can meet the conditions of the problem.

III. Conversely, for this number $b = -2$, according to I., the terms $a_{n}$ (of the sequence defined by (1)) that satisfy the inequalities (2) are those whose index $n$ satisfies both (5) and (6), i.e.,

$$
-14.5 + 20 < n < -24.5 + \frac{100}{3} \quad \rightarrow \quad 5 \frac{1}{2} < n < 8 \frac{5}{6}
$$

i.e., exactly the three terms $a_{n}$ with $n = 6, 7, 8$.

From II. and III., it follows that exactly one integer $b = -2$ satisfies the conditions, and the corresponding three terms $a_{n}$ that satisfy (2) are derived from (1) with (7):

$$
a_{6} = \frac{16}{11} \quad ; \quad a_{7} = \frac{19}{13} \quad ; \quad a_{8} = \frac{22}{15}
$$

Adapted from $[5]$

### 9.28.2 Second Round 1986, Class 12