Summarize the following:
The key to understanding this formula is to recognize the underlying pattern. The sum of any set of consecutive odd numbers starting with 1 is always equal to the square of the number of digits that were added together.  Sum of first odd number = 1 Sum of first two odd numbers = 1 + 3 = 4 (= 2 x 2). Sum of first three odd numbers = 1 + 3 + 5 = 9 (= 3 x 3). Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16 (= 4 x 4). By solving this problem, you learned more than the sum of the numbers. You also learned how many consecutive digits were added together: 41! This is because the number of digits added together is always equal to the square root of the sum.  Sum of first odd number = 1. The square root of 1 is 1, and only one digit was added. Sum of first two odd numbers = 1 + 3 = 4. The square root of 4 is 2, and two digits were added. Sum of first three odd numbers = 1 + 3 + 5 = 9. The square root of 9 is 3, and three digits were added. Sum of first four odd numbers = 1 + 3 + 5 + 7 = 16. The square root of 16 is 4, and four digits were added. Once you understand the formula and how it works, you can write it down in a format that will be applicable no matter what numbers you are dealing with. The formula to find the sum of the first n odd numbers is n x n or n squared.  For example, if you plugged 41 in for n, you would have 41 x 41, or 1681, which is equal to the sum of the first 41 odd numbers. If you don't know how many numbers you are dealing with, the formula to determine the sum between 1 and n is (1/2(n + 1))2

summary: Observe the pattern. Understand the interim data. Generalize the formula.


Summarize the following:
If you've written a book, you may be wondering what kind of residual income you can earn from the royalties. The specifics will depend on the contract you agree to sign with the publishers, so it's important to be aware of what rights you can and can't invoke during this process.  Only give your publisher the specific rights that group will need for the book at hand. For example, if the publisher primarily works with English-language publications in the United States, you should be skeptical if that publisher tries to secure international rights to your book. Grant the publisher subsidiary rights that relate only to the sale of the book itself (for example, large-print rights or book club rights). Any outside rights your publisher tries to secure, like movie or TV rights, should be off the table. Have your agent argue for very specific language that details the terms of your royalties. All royalties in your contract should be a specified percentage of the suggested retail price of the book, not on the publisher's net receipts. Any attempt to stray from royalties derived from the suggested retail price will only cause you to lose money to the publisher. If you negotiate for an advance on the book, do not let the publisher issue payment "upon execution" (or similarly worded specifications). Make sure the advance is received as soon as your contract is delivered to the publisher, or specify a certain number of days after your contract is received in which the check must be issued. Make sure that your contract requires the publisher to have your book registered for copyright within 90 days of publication to avoid any risk of copyright infringement or intellectual theft. If you're a musician and you've written a successful song, you may be able to earn royalties from the sale of that song. Once you've gotten a record contract offer, you'll need to review the terms of that contract to ensure that you receive your fair share of the royalties.  As the artist who wrote/performed a song, you own the rights to the master recording, and you assign rights to your music label in order for them to distribute and promote your song(s). Some labels may try to acquire full ownership over a recording's copyright. It's in your best interest, financially and creatively, to maintain control by keeping the copyright and only giving rights to the label for a limited period of time (often called a license deal). Never sign a record agreement that asks you to transfer copyrights or to take on a work-for-hire role. This is a bad contract that will exploit you and your work. Make sure your contract specifies both digital and physical royalties, ideally based on net published price to dealer (wholesale price after distribution and merchant fees). That way you will earn royalties off of digital downloads and physical record/CD/cassette sales. Radio play royalties are split across three categories: commercial radio, classical radio, and college radio. Hit songs and long-standing "standards" are often eligible for bonus royalties, depending on the terms of your contract. Another residual income option that some people might be interested in is peer-to-peer lending. In peer-to-peer (P2P) lending, you essentially provide someone a loan just like a bank or credit union would. The returns you get from a P2P loan are far greater than the interest you would receive on an average savings account, CD, or bond.  Always go through a trustworthy P2P site, like Lending Club or Prosper. The P2P site will take care of running a credit and income check on potential borrowers, then match you with qualifying borrowers who pass those checks. Make sure that the P2P site you're considering will service the loans and manage the collection process. To get started, you'll simply open an online account and make your funds available. Then choose the terms of the loan (usually either 36 months or 60 months), the credit risk and projected return you feel comfortable with, and the amount you'd like to invest in each lender.  If you're unsure about the terms of a loan, consider talking to a financial planner or accountant who can walk you through the lending process and explain the risks vs. rewards of each option you have. . In addition to selling items or services and participating in affiliate marketing, you can use your website or blog to make money by selling advertising space. If you have a highly-trafficked website and/or highly-involved regular readers, you have a way to attract advertisers for related products and have them purchase advertising space on your site. However, this advertising will have to be related to your website or blog in some way, otherwise your readers won't be interested.  Use the information in the method "Earning Income Through Affiliate Marketing" in this article to build a website or blog and a devoted readership. You may have to redesign your site or blog to accommodate advertising space. Try experimenting with different layouts so that you can bring in ads without losing the professional feel of your site. Most online advertising is cost per click (CPC) advertising. This means that the advertiser pays you a small amount each time their ad on your site is clicked on. Mobile applications are a large and still-growing market that can provide you with a large amount of income if you know how to capitalize on it. The first and most challenging step, however, is creating an app that people actually want to use. Try to assess the need for an app by thinking of useful apps you would want or problems that can be solved by an app. Then, look for current apps that met these needs. If they aren't any, or if they aren't any quality ones, consider creating your own app.  After you create your app, you will need a way to monetize it. This can be done in several ways. For one, you can include in-app advertising, like banners that show up during use of the app. Another common method for monetizing apps is in in-app purchases. These add additional features to the app, like unlocking more levels in a game or adding filters to a photo editing app. Other apps make money simply by charging users to download the app. Finally, e-commerce apps allow users to purchase items or services through the app.
summary: Earn book royalties. Get royalties from a song. Try peer-to-peer lending. Generate online advertising revenue Create and monetize an app.