#Series and sequences series

The method used to find the general term given the summation formula for terms is, This not only applies to APвЂ™s but to all series. Now, in some questions, students are given the summation formula, and are required to find the general term of the sequence being summed. What is the sum of the first 80 terms of the sequence ? Solution 6 Otherwise we simply use the standard form which is the first.

Note: The second form of the summation formula for an AP is only used if the last term is known. Where is the first term, is the number of terms in the series, is the last term and is the common difference. Now, since there exists terms in the series, we thus have that where is the last term. Which is the formula for the summation of an AP.

#Series and sequences series

Now, adding the two series with terms lining vertically being added gives,

Now, we consider written in the reverse order with terms lining up vertically with written in correct order. Now, suppose we are adding terms from an arithmetic progression to obtain a sum. We denote the sum up to terms for any series as. We now shall look at arithmetic series (the summation of an arithmetic progression). Thus we have that the first term is and the common difference is for this sequence. We firstly have that the general term of an AP is given by. Find the first term and the common difference of this sequence. The rd term of an AP is and the st term is.

Hence we have that the rd term is equal to. So we must set the general term to and solve the resulting equation in to find the term number. Solution 4ДЇirstly, we find the general term. Hence we have that the general term is given by,ДЇind the term number that is equal to in the sequence. We shall now consider some examples to illustrate use of this formula. Where is the first term, is the common difference and is the term number. Then we have the following table.Д«y observation of the general pattern, we can observe that the th term of an arithmetic sequence is given by Suppose that the first term of an arithmetic progression is and that the common difference is given by. We shall now find the general term describing the th term of any arithmetic progression. Hence as the difference is constant, then it follows that the terms form an arithmetic progression.ДЄs a convention, we denote the common difference between successive terms of an AP by.

That is, each successive pair, differ by the same amount. Solution 2ДЇor three successive terms to form an arithmetic progression, we simply require that the difference between the third and second term be equal to the difference between the second and first term. Show that forms an arithmetic progression. For example, consider the arithmetic progression all the adjacent terms of which differ by. In an arithmetic progression, ensuing terms follow the pattern of differing by the same amount. Arithmetic SequencesДЄ special family of sequences called arithmetic sequences (or arithmetic progressions abbreviated to AP), are of particular interest to us. That is, the general rule, is simply a function which converts natural numbers into a sequence. This is because a sequence is in fact simply the range of a function with a domain equal to the natural numbers. Note: The expression is in fact a short hand convention for. Hence we have that the first three terms are given by and respectively. Write down the first three terms for the sequence given by the rule.