Article: .. given by the recursion shown. This cannot be solved by any of the above methods, but a formula can be found by using generating functions. A generating function is simply a formal power series where the coefficient of xn is the nth term of the sequence. The objective in this step is to find an equation that will allow us to solve for the generating function A(x). Extract the initial term. Apply the recurrence relation to the remaining terms.  Split the sum. Extract constant terms. Use the definition of A(x). Use the formula for the sum of a geometric series.  The methods for doing this will vary depending on exactly what A(x) looks like, but the method of partial fractions, combined with knowing the generating function of a geometric sequence, works here as shown.
Question: What is a summary of what this article is about?
Consider the sequence 2, 5, 14, 41, 122 . Write the generating function of the sequence. Manipulate the generating function as shown. Find the generating function A(x). Find the coefficient of the xn in A(x). Write the formula for an by identifying the coefficient of xn in A(x).

Problem: Article: If your pimple is red and inflamed but not ready to pop, a hydrocolloid dressing can help calm it quickly. Put the hydrocolloid bandage on your pimple overnight to minimize inflammation and help absorb excess oil and pus.  Hydrocolloid dressings are the same type of rubbery bandage that are sometimes used to treat blisters. You can find them in your local pharmacy or online. This treatment won’t heal your pimple overnight, but it will make it a little less obvious. Use the treatment for a few days to help your pimple heal up completely. Acne spot treatments can help dry out pimples and reduce inflammation to promote faster healing. They are also formulated with medications that help kill the germs that cause pimples to form. While these treatments won’t heal a pimple instantly, they can help reduce its appearance and encourage it to heal up more quickly. Look for acne spot treatments containing ingredients such as:  Benzoyl peroxide Salicylic acid Sulfur While it won’t actually get rid of your pimple, a little concealer or coverup can make a pimple a lot less noticeable. Dab the concealer on the spot and gently pat it to blend it in with the surrounding skin.  A concealer with a green tint can help offset the redness of the pimple. Before applying the concealer, wash the affected area to remove grease, dirt, and germs. You can also make the concealer more effective by combining it with other treatments. For example, use a hydrocolloid patch overnight to reduce inflammation, then put on the concealer in the morning. It can take weeks for a pimple to surface and become ready to drain. If you just can’t wait that long, a cortisone injection can help it heal up in a matter of hours to a couple days. Talk to your doctor about getting a cortisone treatment, or ask for a referral to a dermatologist who can do it. A dermatologist can also recommend treatments to prevent future outbreaks.
Summary: Use a hydrocolloid dressing to reduce swelling overnight. Try a spot treatment to dry out your pimple. Hide obvious pimples with concealer. See a doctor for a cortisone injection if you need a quick fix.

Like all dogs, Belgian Malinois must be taught to appropriately greet and interact with other dogs and people. You should make a conscious effort to expose your dog to new environments, experiences, people, and animals. Belgian Malinois are a social breed and enjoy interacting with other dogs and people. Since they are very protective, they may initially be wary of strangers. By exposing them to new people in safe environments, they learn that new people are not a threat. Introduce your dog to your neighbors, friends, and family. Use a friendly tone of voice and be excited to see the new person. This lets the Malinois know that you do not perceive these new people as a threat. Once they realize they are not a threat, let the dog approach the new person. Give your friend a few treats to share with the dog so they can form positive associations with meeting new people. Belgian Malinois do enjoy playing with other dogs and it is important to socialize your dog to interact positively with other animals. Take your Malinois to areas where other dogs are, like dog day cares or on walks with other dogs. This gives them an opportunity to play with a variety of other dogs and to be exposed to new environments. If your friends and family have dogs, set up a play date so the two dogs can meet. Let the dogs approach each other and become acquainted. Demonstrate that you do not perceive the other dog as a threat to your safety by petting it and speaking in a kind and reassuring tone. Belgian Malinois want to please their owners by following their directions and commands. They are working dogs that are very task oriented and enjoy being rewarded for their hard work.  Small treats can be very helpful in training exercises, like in teaching obedience commands. Reward your Belgian Malinois immediately after they complete the task you have instructed them so they associate your praise with the action.  Positive reinforcement, through treats or praise, can also help condition your dog to respond favorably to new situations. Belgian Malinois are a very intelligent breed and will learn new tasks and tricks easily. They are bred to be working dogs so they have a strong drive to follow directions and carry out tasks.  Basic commands like sit, stay, and come are all good first commands to begin with.  Teach your Malinois to sit by showing them a treat. When the dog looks up at the treat, move the treat close to the dog's nose, then slowly back over the dog's head. When the dog tries to follow the treat with its eyes and nose, it will usually sit down. When this happens, give the dog the treat and say "Sit." Stay and come can be taught in tandem. Have the dog sit in front of you while you are standing. Take a few steps backwards as you say, “stay.” If the dog gets up to follow you, instruct them to sit again and repeat the process. When the dog has learned to stay, you can begin to use the “come” command to instruct them to follow you as you back away. your dog. Belgian Malinois tend to be very attached to their owners and families, so there is less of a risk of them running away than with other breeds. However, they do need to be taught to walk on a leash and to follow your commands.  Belgian Malinois are very strong and can pull while on a leash. If your dog begins to pull while you are walking them on a leash, you can stand still and wait until they reach the end of the leash and are forced to walk back to you. You can also use treats to encourage dogs to stay close to your side and to match your gait.  Do not use retractable leashes. They are often not strong enough to be used with larger dogs, and they reinforce bad leash habits like walking far ahead or behind you. They can also negatively impact the socialization work that you’ve already done. If you retract the leash anytime another dog or person comes close to you, you are signaling to your Malinois that you perceive that person or dog to be a threat. When it comes to training, practice makes perfect. You will need to spend time each day - depending on your dog’s obedience level it may be numerous 15 minute intervals - practicing the commands and tricks that you want the dog to learn.  Most Belgian Malinois will enjoy training since it gives them an opportunity to spend time with you and provides mental stimulation. Once your dog is fully trained, you should continue to frequently practice the commands that they have learned so they retain them over time. While most Belgian Malinois are easy to train, some owners feel overwhelmed with the training process if they have not trained a dog before. Visit your local pet store for information on obedience classes, or search for classes in your area online.  There are classes for puppies and adult dogs, and most will allow you to sit in on the classes so you learn proper techniques. You can also enroll in an obedience class to teach your Malinois more advanced commands.
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One-sentence summary --
Introduce your dog to new environments. Let your dog engage with other dogs. Use positive reinforcement. Teach basic obedience commands. Leash train Train consistently. Enroll in an obedience class.