sec 9.2 Arithmetic Sequences Definition of Arithmetic Sequence An ...

sec 9.2 Arithmetic Sequences Definition of Arithmetic Sequence An arithmetic sequence is a sequence of the form a, a  d, a  2d, a  3d, a  4d, . . . The number a is the first term, and d is the common difference of the sequence. The difference between 2 consecutive terms is d a n − a n−1  d —————————— The nth Term of an Arithmetic Sequence The nth term of the arithmetic sequence a, a  d, a  2d, a  3d, a  4d, . . . is a n  a  n − 1d ————————— Example 1 a. Find the arithmetic sequence if a  2 and d  3 a n − a n−1  d or a n  a n−1  d 2, 2  d  5, 5  d  8, 8  d  11 adding d to the previous term to get the next sequence is 2,5,8,11.... b. Consider the arithmetic sequence 9,4,-1,-6,-11.... Find the nth term a n  a  n − 1d and a n − a n−1  d d2nd term-1st term  4 − 9  −5 a1st term  9 a n  a  n − 1d a n  9  n − 1−5  14 − 5n do Now c. Find a 100 using a n  a  n − 1d ans: -486 Example2 Find the first 6 terms and the 300th term of the arithmetic sequence 13,7.... a n  a  n − 1d and a n − a n−1  d d2nd - 1st term 7 − 13  −6 13,7,1,-5,-11,-17... adding d to the previous term to get the next ———————The 300th term a n  a  n − 1d

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n  300, d  −6, a  13 a 300  13  300 − 1−6  − 1781 —————————– Do NOW Given a  3, d  5 a. find the first 4 terms of the arithmetic sequence b. find the 100th term ans. 3,8,13,18,....b. a 100  3  100 − 15  498 —————————Example3 Finding terms in the arithmetic sequence Given the 11th term of an arithmetic sequence is 52 and the 19th term is 92, find the 1000th term ———————————————a n  a  n − 1d a 11  a  11 − 1d 52  a  10d a 19  a  19 − 1d 92  a  18d Solve 2 equations in 2 unknowns 52  a  10d 92  a  18d subtract ——————– −40  −8d d5 Substitute into 1st equation for d 52  a  105 Solve for a 52  a  50 a2 ———————The 1000th term a n  a  n − 1d n  1000, d  5, a  2 a 1000  2  1000 − 15  4997 ———————– Do NOW 1.The 10th term of an arithmetic sequence is 55 and the 2nd term is 7. Find the 1st term ———————— 2. The 12th term of the arithmetic sequence is 32 and the 5th term is 18 Find the 20th term ans. a n  a  n − 1d a 10  a  10 − 1d 55  a  9d a 2  a  2 − 1d 7  ad 2

Solve 2 equations in 2 unknowns 55  a  9d 7  a  d subtract ——————– 48  8d d6 Substitute into 2nd equation for d 7  ad 7  a6 Solve for a  1 ———————————— 2. a10,d2,a 20 48 ———————————— Partial sum of Arithmetic Sequence For arithmetic sequence a n  a  n − 1d The nth partial sum S n  a  a  d  a  2d . . a  n − 1d  1. S n  n 2a  n − 1d or 2 2. S n  n a  a n  2 ——————————— Example 4 Find the sum of the first 40 terms of the arithmetic sequence 3,7,11,15... ———– Use 2. S n  n a  a n  2 d4,a3,n40 a 40  3  40 − 14  159 a n  a  n − 1d a 40  159 S n  n a  a n  2 S 40  40 3  159  20162  3240 2 —————————– #40 Find the partial sum of the arithmetic sequence a3,d2,n12 ———— Use 1. S n  n 2a  n − 1d 2 S 12  12 2  3  12 − 12 2 S 12  66  11  2  6  28  168 ———————– #37 Given seq 1,4,7...

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Which term is 88? ————————find n a1,d3 a n  a  n − 1d a n  1  n − 13  88 Solve for n 1  n − 13  88, Solution is: 30 —————— Do #35 NOW

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