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Algebra 2 AII.2 Sequences and Series Notes Mrs. Grieser Name: _____________________________________ Date: _______________ Block: _______ Sequences and Series Definitions: sequence an ordered list of numbers term each number in a sequence series the sum of a sequence Arithmetic Sequences Domain of a sequence: natural numbers {1, 2, 3, …} Range: value of the terms in the sequence Some sequences show patterns; some do not Finite sequences contain a finite (countable) number of terms Infinite sequences contain an infinite (uncountable) number of terms Arithmetic sequences contain a pattern where a fixed amount is added from one term to the next (common difference d) after the first term Geometric sequences contain a pattern where a fixed amount is multiplied from one term to the next (common ratio r) after the first term Arithmetic sequence examples: o 1, 4, 7, 10, 13, 16, … o Domain: _______________________ o Range: ________________________ o Graph shown at right o common difference d = _________ o The graph of an arithmetic sequence is ______________ o Find the common difference (d) for the following arithmetic sequences: a) 15, 10, 5, 0, … d = _______ b) 1, Write the next 2 terms of each sequence: a) ______________ b) _____________ Write the 8th term of each sequence: 1 1 , 0 , - , …. d = _______ 2 2 a) ______________ b) _____________ Algebra 2 AII.2 Sequences and Series Notes Mrs. Grieser Page 2 Notation: o an = nth term of an arithmetic sequence o a1, a2, …, an : terms of an arithmetic sequence Finding Terms in a Sequence o Find the 8th term in the sequence: 5, 9, 13, 17, … d = _____________ a8 = ____________ o Is there a pattern? a1 = 5 a2 = a1 + d a3 = a2 + d = a1 + d + d = ___________ a4 = a3 + d = a2 + d + d = a1 + d + d + d = ___________ an = _____________ To find the nth term in an arithmetic sequence: an = a1 + (n – 1)d where a1 is the first term of the sequence, d is the common difference, n is the number of the term to find o You try…Find the requested term in the sequence: a) Find the 7th term: 3, 9, 15, 21, … b) Find the 10th term: 3, 5, 7, 9, … c) Find a7: a1 = 3x and d = -x d) If a4 = 6 and d = -2, find a5 Sequence Formulas o Rather than writing out terms, we often use a formula or rule for a sequence o Example: an = 2n o Write out the first 6 terms: ___________________________________ o What is a10? ___________ o Example: Given an = 6n + 3 Find the first 5 terms _______________________________ What is the common difference d? ________________ What is the 10th term of the sequence? ____________ What is a15? ___________ Algebra 2 AII.2 Sequences and Series Notes Mrs. Grieser Page 3 o Example: Find a formula for the sequence 1, 3, 5, 7, … General formula for the nth term: _____________ What is the common difference? ________ Substitute common difference for d, and simplify _________________ o Example: One term of an arithmetic sequence is a19 = 48. The common difference is d = 3. Use the general rule for the nth term: _____________________ o Example: Insert three arithmetic means between 7 and 23 An arithmetic mean is the term between any two terms of an arithmetic sequence. It is simply the average (mean) of the given term. Use the general formula for the nth term to find d, the common difference: Now use d to find 3 arithmetic means: ________________________ o You Try…Answer each question, then find a20 for each sequence. a) Find the first 6 terms of the sequence: an = 6 - n b) Write a rule for the sequence given a11=-57 and d = -7 c) Write a rule for the sequence that has a7=26 and a16=71 Arithmetic Series An arithmetic series is the sum of an arithmetic sequence: Sn= n a i 1 Summation Notation Review o Read: the sum as i goes from 1 to infinity of ai o Can be finite or infinite o index starts at lower limit of summation, then increments by 1 until upper limit is reached o Index doesn’t have to be i; can be any letter o Index doesn’t have to start at 1 i Algebra 2 AII.2 Sequences and Series Notes Mrs. Grieser Page 4 o Examples: 4 2i a) 2i i 1 b) i 1 5 (m 2 1) c) m 1 o Sum of a constant: n k 6 4 1 d) i 5 e) i 1 j 1 k 2 k 1 = _____________ i 1 Find 12 5 ______________ i 1 o Sum of the first n numbers: n i = __________ i 1 Find 50 k ______________ k 1 o Sum of the squares of the first n numbers: n i 2 = __________ i 1 Find 18 k 2 ______________ k 1 o Other properties of summation: n n i 1 i 1 kai k ai , where k is a constant 10 4i Find i 1 n n n ( a b ) a b i i 1 Find i i 1 i i 1 i 8 (i 7) i 1 o You Try…Find the values of the summations: a) 7 8i i 1 b) 12 (k 10) k 1 10 c) (k 2 1) k 1 20 d) ( j j 1 2 j 5) Algebra 2 AII.2 Sequences and Series Notes Mrs. Grieser Page 5 Sums of a Finite Arithmetic Series o The sum of the first n terms of an arithmetic series is n times the mean of the first and last terms: a a Sn = n 1 n 2 Find 10 (3i 1) i 1 n = _____, a1 = _______, an = _______ Use formula: You try: Find 20 (3 5i) i 1 n = _____, a1 = _______, an = _______ Use formula: o What if you don’t know an, but you do know the common difference, d? a an Sn = n 1 2 an = a1 + (n-1)d a an a a1 (n 1)d 2a (n 1)d Substitute: Sn = n 1 = n 1 n 1 2 2 2 o Example: Find the sum of the first 15 terms of the sequence 3, 7, 11, … n = _____________ a1 = _______________ d = _______________ Use formula: Sum: ___________ o You Try… a) Find 12 (2 7k ) b) Find the sum of the first 10 terms of the sequence: 9, 5, 1, … k 1 c) During a high school spirit week, students dress up in costumes, with a cash prize being given each day for the best costume. The organizing committee has $1,000 to give away over 5 days. The committee wants to increase the amount by $50 each day. How much should the committee give away on the first day?