Arithmetic Sequences: Finding Terms and Common Differences, Summaries of Mathematics

Download Arithmetic Sequences: Finding Terms and Common Differences and more Summaries Mathematics in PDF only on Docsity! Mathematics Arithmetic Sequences Science and Mathematics Education Research Group Supported by UBC Teaching and Learning Enhancement Fund 2012-2013 Department of Curr iculum and Pedagogy F A C U L T Y O F E D U C A T I O N a place of mind Arithmetic Sequences | a oe a a A. a8 = a5 + 3d B. a8 = a5 + 3a1 C. a8 = a5 + 8d D. a8 = a5 + 8a1 E. Cannot be determined Arithmetic Sequences II Consider the following sequence of numbers: a1, a2, a3, a4, a5, ... where an is the nth term of the sequence. The common difference between two consecutive terms is d. What is a8, in terms of a5 and d? Solution Answer: A Justification: The next term in the sequence can be found by adding the common difference to the last term: a1, a2, a3, a4, a5, a6, a7, a8 Only 3 times the common difference has to be added to the 5th term to reach the 8th term. Notice that the first term does not need to be known. As we will see in later questions, it will be helpful to be able to express terms of a sequence with respect to the first term. +d +d +d a8 = a5 + d + d + d = a5 + 3d A. a8 = 8a1 B. a8 = a1 + 6d C. a8 = a1 + 7d D. a8 = a1 + 8d E. Cannot be determined Arithmetic Sequences III Consider the following sequence of numbers: a1, a2, a3, a4, a5, ... where an is the nth term of the sequence. The common difference between two consecutive terms is d. What is a8, in terms of a1 and d? Solution Answer: D Justification: Consider the value of the first few terms: a1 = a1 + 0d a2 = a1 + 1d a3 = a1 + 2d a4 = a1 + 3d ⋮ an = a1 + (n-1)d Notice that the common difference is added to a1 (n-1) times, not n times. This is because the common difference is not added to a1 to get the first term. Also note that the first term remains fixed and we do not add multiples of it to find later terms. A. a21 = 6 + 20(5) B. a21 = 21 + 20(5) C. a21 = 21 + 21(5) D. a21 = 21 – 20(5) E. a21 = 21 – 21(5) Arithmetic Sequences V Consider the following arithmetic sequence: __, __, __, 6, 1, ... What is the 21st term in the sequence? Press for hint Hint: Find the value of the common difference and the first term. an = a1 + (n-1)d Solution Answer: D Justification: The common difference is d = a5 – a4 = 1 – 6 = -5. Subtracting the common difference from an gives an-1. This gives a1 = 21. Using the formula, an = a1 + (n-1)d, we find that: a21 = 21 + (21-1)(-5) = 21 – 20(5) = -79 Arithmetic Sequences VII In a particular arithmetic sequence: a19 = 50, a30 = 80 What is the common difference of this sequence? above the of NoneE. D. C. B. A. 12 30 11 30 10 30 9 30     d d d d Solution Answer: D Justification: (Method 2): Using the formulas, a19 and a30 in terms of a1 is given by: a19 = a1 + 18d a30 = a1 + 29d Subtracting a30 from a19 gives: 11 30 d 11d30 11daa 1930    (Method 1): To get to a30 from a19, 11 times the common difference must be added to a19: 11 30 d 11d30 11daa 11daa 1930 1930     A. a1 = 10; d = 2 B. a1 = 15; d = -3 C. a11 = 30; a12 = 20 D. a20 = 40; d = 2 E. a20 = 40; d = -3 Arithmetic Sequences VIII The statements A through E shown below each describe an arithmetic sequence. In which of the arithmetic sequences is the value of a10 the largest?