Tutor profile: Jessica R.
What are sequences and series, and how are they different?
A sequence is simply a list of numbers. Often, the sequence will be listed as a function, where the x input will determine the y output. In this case, each y output would become a number in the sequence. A series, however, is a summation of a sequence. For example, if the sequence is: 1, 2, 3, 4, 5, then the corresponding series would be: 1 + 2 + 3 + 4 + 5. A series is merely the sum of all the numbers in a sequence. When we involve functions in our sequences, the series can get slightly more complicated, but the basic principle is always the same.
Long essays: how do you write an analysis in a long essay or document-based question, especially when it is an in-class essay with a time constraint?
The most important part about writing an essay is to be able to look at your essay with the perspective of the reader. When we write essays, it's very easy for us to skip a step in the analysis. After all, our analysis usually makes perfect sense in our head, which often causes us to assume it will also make perfect sense for the reader. However, the reader of our essay may not follow our same train of thought and they might get lost in an incomplete analysis. It is important to be able to read through an analysis as if you have no knowledge on the subject, and see if it still makes sense. More often than not, the solution is to break down the analysis step by step, connect ALL the dots (even the ones that seem obvious), and give the reader no choice but to understand and follow your argument. An easy way to do this is to ensure that every point you make ultimately relates back to your original thesis. Often, this is as easy as adding an extra sentence to explain exactly how and why the statement is important to your claim, even if you think it should be obvious. A clear and explicit analysis is the key for a successful long essay.
Understanding exponents: Why is anything, raised to the power of 0, equal to 1?
In order to understand this, we have to look at exponents as a pattern. Let's take the number 2. 2 raised to the first power equals 2; 2 raised to the second power equals 4; and 2 raised to the third power equals 8. As we increase the power by 1, the answer yielded is doubled, or multiplied by 2 -- which is what makes the pattern exponential. Therefore, we can look at this very same pattern, but backwards. As we decrease the power (from third, to second, to first), the answer yielded will be cut in half, or divided by 2 (it goes from 8 to 4 to 2). So if we decrease the power again, from the first power to the zero power, it only makes sense to continue to divide the answer by 2. 2 raised to the first power equals 2, and therefore 2 raised to the zero power equals 1 (which is half of 2). We can use this pattern to determine the solutions for negative exponents. Also, we can use this pattern to prove this theory for any number (but instead of multiplying and dividing by 2, we would multiply/divide by the whatever number we are using).
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