How to calculate this value? It's easy - all we have to do is subtract the distance traveled in the first four seconds, S₄, from the partial sum S₉. However, we're only interested in the distance covered from the fifth until the ninth second. S₉ = n/2 × = 9/2 × = 388.8 mĭuring the first nine seconds, the stone travels a total of 388.8 m.
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The distance traveled follows an arithmetic progression with an initial value a = 4 m and a common difference, d = 9.8 m.įirst, we're going to find the total distance traveled in the first nine seconds of the free fall by calculating the partial sum S₉ ( n = 9): What is the distance traveled by the stone between the fifth and ninth second? Every next second, the distance it falls is 9.8 meters longer. During the first second, it travels four meters down. We will take a close look at the example of free fall.Ī stone is falling freely down a deep shaft. Let's analyze a simple example that can be solved using the arithmetic sequence formula. This formula will allow you to find the sum of an arithmetic sequence. Substituting the arithmetic sequence equation for nᵗʰ term: All you have to do is to add the first and last term of the sequence and multiply that sum by the number of pairs (i.e., by n/2). That means that we don't have to add all numbers. The sum of each pair is constant and equal to 24. We will add the first and last term together, then the second and second-to-last, third and third-to-last, etc. Let's try to sum the terms in a more organized fashion. We could sum all of the terms by hand, but it is not necessary. Look at the first example of an arithmetic sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. Trust us, you can do it by yourself - it's not that hard! Our arithmetic sequence calculator can also find the sum of the sequence (called the arithmetic series) for you. A perfect spiral - just like this one! (Credit: Wikimedia.) If you drew squares with sides of length equal to the consecutive terms of this sequence, you'd obtain a perfect spiral. It's worth your time.Ī great application of the Fibonacci sequence is constructing a spiral. Interesting, isn't it? So if you want to know more, check out the fibonacci calculator. Each term is found by adding up the two terms before it.
This is not an example of an arithmetic sequence, but a special case called the Fibonacci sequence. Now, let's take a close look at this sequence:Ĭan you deduce what is the common difference in this case? What happens in the case of zero difference? Well, you will obtain a monotone sequence, where each term is equal to the previous one.
Naturally, if the difference is negative, the sequence will be decreasing. If the common difference of an arithmetic sequence is positive, we call it an increasing sequence. In fact, it doesn't even have to be positive! Some examples of an arithmetic sequence include:Ĭan you find the common difference of each of these sequences? Hint: try subtracting a term from the following term.īased on these examples of arithmetic sequences, you can observe that the common difference doesn't need to be a natural number - it could be a fraction.
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