![]() ![]() The n in the formula represents the term number, so you can substitute any whole number in place of n to find the value of the corresponding term. This formula allows you to calculate the value of any term in the sequence, given the value of the first term, a_0, and the common difference, d. The nth term of an arithmetic sequence with a common difference d is denoted by a_n, and it is given by the formula a_n = a_0 + dn, where a_0 is the initial value of the sequence. Source: Online Math Learning □ Formula and Example It's important to note that arithmetic sequences can be negative or positive, if the common difference is negative, it means the terms are decreasing and if the common difference is positive, it means the terms are increasing. The rate of change between the terms is 3 and it is consistent throughout the sequence! Phew… □įor example, an arithmetic sequence with a first term of 5 and a common difference of 3 will look like this: 5, 8, 11, 14, 17, 20. Therefore, we can say that an arithmetic sequence is a function where the input variable is the term number, and the output variable is the value of the term, and the rate of change is constant. The rate of change is the common difference, d. ➕īecause the difference between any two consecutive terms is the same, the terms in an arithmetic sequence have a constant rate of change. , where the nth term is given by a + (n-1)d. For example, if the first term of an arithmetic sequence is a, and the common difference is d, the sequence can be represented as a, a + d, a + 2d, a + 3d. This constant difference is called the common difference. ➖ Arithmetic SequencesĪn arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. □Īgain, we’re looking at discrete points-don’t connect the dots like you’d typically do to draw a curve! Note that the sequence above is just a regular ol’ sequence (not an arithmetic or geometric sequence). If we graph this, we would get a set of discrete points on a graph, one for each day of the week, instead of a smooth curve. ![]() This is a sequence, where the input variable is the day of the week and the output variable is the number of hours of sleep. You start on Monday and get 8 hours of sleep, on Tuesday you get 6 hours of sleep, on Wednesday you get 7 hours of sleep and so on. □įor example, you might want to track the number of hours of sleep you get each night. So, instead of having a smooth curve, we have a set of distinct points on a graph. This is because in a sequence, we are only dealing with whole numbers, not with a continuous range of numbers like in a function. ![]() □Ĭonsequently, the graph of a sequence consists of discrete points instead of a curve. Each day of the week is assigned a number, Monday is 1, Tuesday is 2, and so on. For example, think of a sequence of your class schedule, Monday is the first day, Tuesday the second, and so on. ![]() This means that, a sequence takes a whole number (say, " n") and assigns it a real number. That has saved us all a lot of trouble! Thank you Leonardo.įibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence.A sequence is a function from the whole numbers to the real numbers. "Fibonacci" was his nickname, which roughly means "Son of Bonacci".Īs well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). His real name was Leonardo Pisano Bogollo, and he lived between 11 in Italy. Historyįibonacci was not the first to know about the sequence, it was known in India hundreds of years before! Which says that term "−n" is equal to (−1) n+1 times term "n", and the value (−1) n+1 neatly makes the correct +1, −1, +1, −1. In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+. (Prove to yourself that each number is found by adding up the two numbers before it!) ![]()
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