![]() The sum of the first 12 positive multiples of 5. ![]() Using either of these formulae, calculate: The formula is most useful when you have the first term, the common difference, and the number of terms. That is, the formula is most useful when you have the first term, the last term, and the number of terms. The two formula can be used interchangeably, however you can save yourself one line of work if you use the formula that requires the information you have. We can pick out the term from the formula, and simply replace it with its calculation. We know that can be calculated with the formula. Notice, in our formula, we see the value ‘ ‘. Which can be remembered as ‘first term plus last term, multiply by the number of terms, divided by 2’. Now as the left hand side is double the number we require, let’s divide by 2:įinally, let’s replace the number of terms – 8 in our example – with. Write the sequence a second time and reverse the order of terms:Īdding the terms pairwise, we see that we get eight numbers that are equal. But with Gauss in the room, the task was over in seconds! The website has a nicely written article on this story. His teacher had given the class the task, expecting it to take some time. While still a primary school child, he impressed his teachers by calculating that. We calculate that double the sum of the first seven terms is:ĭivide both sides by 2 to calculate the sum required:Ī mathematician named Karl Frederich Gauss has been called ‘the prince of mathematics’ – due to his many contributions to mathematics. We see that the height of the rectangle is the number of terms in the sequence, in this case 7, and the base of the rectangle is the first term plus the last term. Now we create an area that is double the required sum. The second shape is congruent to the first and fits like a puzzle to form a larger rectangle: Fit two trapezoids together, then divide by 2Īnother way to consider this sum is to create a second trapeziod-like shape with the terms of the sequence. The height is the number of terms in the sequence. Notice that that the base of the rectangle is calculated as the average of the first and the last term. Move the slider to see how the trapezoid-type shape rearranges to a rectangle. Linear means straight.The trapezoid-type shape below is made from the terms of an arithmetic sequence. If we represented an arithmetic sequence on a graph it would form a straight line as it goes up (or down) by the same amount each time. Here are some examples of arithmetic sequences:Īrithmetic sequences are also known as linear sequences. In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic. ![]() The difference between consecutive terms, an an 1, is d, the common difference, for n greater than or equal to two. The term-to-term rule tells us how we get from one term to the next. An arithmetic sequence is a sequence where the difference between consecutive terms is always the same. If we add or subtract by the same number each time to make the sequence, it is an arithmetic sequence. The difference between consecutive terms is an arithmetic sequence is always the same. General expression of arithmetic sequence a, a + d, a + 2d, a + 3d The general term, i.e., nth term in an arithmetic sequence is given by: Formula 2: an a. An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term.įor example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6.Īn arithmetic sequence can be known as an arithmetic progression.
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