answer:1. **Define the sequence and the auxiliary sequence:** Given the sequence ( {a_n} ) defined by: [ a_{n+1} = text{lcm}(a_n, a_{n-1}) - text{lcm}(a_{n-1}, a_{n-2}) quad text{for} quad n geq 3 ] we introduce an auxiliary sequence ( {b_n} ) for ( n geq 5 ): [ b_n = frac{a_n}{text{lcm}(a_{n-2}, a_{n-3})} ] 2. **Show divisibility properties:** We need to show that ( a_{n-2} mid a_n ) for ( n geq 4 ). This follows from the definition of ( a_{n+1} ): [ a_{n+1} = text{lcm}(a_n, a_{n-1}) - text{lcm}(a_{n-1}, a_{n-2}) ] Since ( text{lcm}(a_n, a_{n-1}) ) is a multiple of ( a_n ) and ( text{lcm}(a_{n-1}, a_{n-2}) ) is a multiple of ( a_{n-2} ), it follows that ( a_{n+1} ) is a multiple of ( a_{n-2} ). 3. **Prove the key claim:** We claim that ( b_{n+1} < b_n ) for ( n geq 5 ). To prove this, consider: [ a_{n+1} = text{lcm}(a_n, a_{n-1}) - text{lcm}(a_{n-1}, a_{n-2}) ] Substituting ( a_n = b_n cdot text{lcm}(a_{n-2}, a_{n-3}) ), we get: [ a_{n+1} = text{lcm}(b_n cdot text{lcm}(a_{n-2}, a_{n-3}), a_{n-1}) - text{lcm}(a_{n-1}, a_{n-2}) ] Since ( a_{n-3} mid a_{n-1} ), we have: [ a_{n+1} = text{lcm}(b_n, a_{n-2}, a_{n-1}) - text{lcm}(a_{n-1}, a_{n-2}) ] Dividing both sides by ( text{lcm}(a_{n-1}, a_{n-2}) ), we get: [ b_{n+1} = frac{a_{n+1}}{text{lcm}(a_{n-1}, a_{n-2})} < b_n ] 4. **Bound ( b_5 ):** To complete the proof, we need to show that ( b_5 leq a_3 - 2 ). Consider: [ a_4 = text{lcm}(a_3, a_2) - text{lcm}(a_2, a_1) = c cdot a_2 ] for some ( c leq a_3 - 1 ). Then: [ a_5 = text{lcm}(a_4, a_3) - text{lcm}(a_3, a_2) = text{lcm}(c cdot a_2, a_3) - text{lcm}(a_3, a_2) ] Dividing by ( text{lcm}(a_3, a_2) ), we get: [ b_5 = frac{a_5}{text{lcm}(a_3, a_2)} leq c - 1 leq a_3 - 2 ] 5. **Conclusion:** Since ( b_n ) is strictly decreasing and ( b_5 leq a_3 - 2 ), there exists some ( k leq a_3 + 3 ) such that ( b_k = 0 ), implying ( a_k = 0 ). (blacksquare)

answer:The evaporation process is a natural phenomenon where a liquid transforms into a gas or vapor. This process occurs when the molecules of a substance gain enough energy to break free from the surface tension of the liquid and turn into vapor. The energy required for evaporation is usually provided by heat from the surroundings, such as the sun or warm air. As the molecules gain energy, they begin to move rapidly and break away from the surface of the liquid, forming a gas. This process is an essential part of the water cycle, where water from the oceans, lakes, and rivers evaporates into the air, eventually forming clouds and leading to precipitation. Evaporation is an endothermic process, meaning it absorbs heat from the surroundings, which is why it often has a cooling effect. For example, when you sweat, the evaporation of sweat from your skin helps to cool you down. The rate of evaporation depends on factors such as temperature, humidity, and wind speed.

answer:The condensation process is the opposite of evaporation, where a gas or vapor transforms back into a liquid. This occurs when the molecules of a substance lose energy and slow down, allowing them to come together and form droplets of liquid. Condensation happens when the air reaches its dew point, which is the temperature at which the air becomes saturated with water vapor and can no longer hold any more moisture. When this occurs, the water vapor in the air condenses onto a surface, forming droplets of water. This process releases heat, which is why condensation is an exothermic process. Examples of condensation include the formation of dew on grass and spider webs in the morning, fog on a cold day, and the water droplets that form on a cold drink on a hot day. In the water cycle, condensation is an essential step, as it allows water vapor in the air to condense into clouds, which eventually produce precipitation. Condensation can also occur through various means, such as through contact with a cold surface or through the presence of condensation nuclei, like dust particles or salt crystals, which provide a site for water droplets to form around.

answer:The precipitation process is the formation and falling of water droplets or ice crystals from the atmosphere to the Earth's surface. This process occurs when the air becomes saturated with water vapor, and the water molecules condense onto tiny particles in the atmosphere, such as dust, salt, or pollutants, forming visible clouds. As more and more water vapor condenses onto these particles, the droplets grow larger and heavier, eventually becoming too heavy to remain suspended in the air. At this point, they fall to the ground as precipitation, which can take various forms, such as rain, snow, sleet, or hail, depending on the temperature and other atmospheric conditions. Precipitation can occur through several mechanisms, including frontal precipitation, where two air masses of different temperatures and humidity levels meet, causing the air to rise and cool, resulting in precipitation. Orographic precipitation occurs when air is forced to rise over mountains, cooling and condensing, resulting in precipitation. Convectional precipitation occurs when warm air rises, cools, and condenses, resulting in precipitation. Once the precipitation reaches the ground, it can either be absorbed into the soil, flow over the surface as runoff, or collect in low-lying areas as standing water. Precipitation is a crucial part of the water cycle, as it helps to distribute freshwater around the planet and replenish the Earth's water sources.