31: Arithmetic Sequences and Series © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

Arithmetic Sequences and Series Module C1 AQA Edexcel OCR MEI/OCR Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

Arithmetic Sequences and Series A sequence is arithmetic if each term – the previous term = d where d is a constant e.g. For the sequence d = 2 nd term – 1 st term = 3 rd term – 2 nd term... = 2 Arithmetic Sequence The 1 st term of an arithmetic sequence is given the letter a.

Arithmetic Sequences and Series Arithmetic Sequence An arithmetic sequence is of the form Notice that the 4 th term has 3 d added so, for example, the 20 th term will be The n th term of an Arithmetic Sequence is An arithmetic sequence is sometimes called an Arithmetic Progression (A.P.)

Arithmetic Sequences and Series Arithmetic Series When the terms of a sequence are added we get a series e.g. The sequence gives the series The Sum of an Arithmetic Series We can derive a formula that can be used for finding the sum of the terms of an arithmetic series

Arithmetic Sequences and Series Arithmetic Series e.g. Find the sum of the 1 st 10 terms of the series Solution: Writing out all 10 terms we have Adding the 1 st and last terms gives 11. Adding the 2 nd and next to last terms gives 11. The 10 terms give 5 pairs of size 11 ( = 55 ). Writing this as a formula we have where l is the last term

Arithmetic Sequences and Series With an odd number of terms, we can’t pair up all the terms. e.g. Arithmetic Series However, still works since we can miss out the middle term giving n = 6. Now we add the middle term We get

Arithmetic Sequences and Series However, still works since we can miss out the middle term With an odd number of terms, we can’t pair up all the terms. e.g. Arithmetic Series Together we have which is giving n = 6. Now we add the middle term which equalsWe get

Arithmetic Sequences and Series For any arithmetic series, the sum of n terms is given by Substituting for l in the formula for the sum gives an alternative form: Since the last term is also the n th term,

Arithmetic Sequences and Series SUMMARY  The sum of n terms of an arithmetic series is given by  An arithmetic sequence is of the form  The n th term is or

Arithmetic Sequences and Series e.g.1Find the 20 th term and the sum of 20 terms of the series: Solution: The series is arithmetic. where Either or

Arithmetic Sequences and Series e.g.2The common difference of an arithmetic series is - 3 and the sum of the first 30 terms is 255. Find the 1 st term. Solution:

Arithmetic Sequences and Series Exercises 1. The 1 st term of an A.P. is 20 and the sum of 16 terms is 280. Find the last term and the common difference. 2. Solution: Find the sum of the series given by We can see the series is arithmetic so, Substituting n = 1, 2 and 3, we get  6,  2, 2

Arithmetic Sequences and Series

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Arithmetic Sequences and Series Arithmetic Sequence An arithmetic sequence is of the form Notice that the 4 th term has 3 d added so, for example, the 20 th term will be The n th term of an Arithmetic Sequence is An arithmetic sequence is sometimes called an Arithmetic Progression (A.P.)

Arithmetic Sequences and Series SUMMARY  The sum of n terms of an arithmetic series is given by  An arithmetic sequence is of the form  The n th term is or

Arithmetic Sequences and Series e.g.1Find the 20 th term and the sum of 20 terms of the series: Solution: The series is arithmetic. whereEither or

Arithmetic Sequences and Series e.g.2The common difference of an arithmetic series is - 3 and the sum of the first 30 terms is 255. Find the 1 st term. Solution: