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Given an arithmetic sequence with a15 38 and d 3, find a1. x a1 First term 38 an nth term 15 n number of terms NA Sn sum of n terms -3 d common difference an a1 n 1 d 38 x 15 1 3 X = 80 What is a Geometric Sequence? In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio. Unlike in an arithmetic sequence, the difference between consecutive terms varies. We look for multiplication to identify geometric sequences. Ex: Determine if the sequence is geometric. If so, identify the common ratio 1, -6, 36, -216 yes. Common ratio=-6 2, 4, 6, 8 no. No common ratio Important Formulas for Geometric Sequence: Recursive Formula an = (an – 1 ) r Where: an is the nth term in the sequence a1 is the first term n is the number of the term r is the common ratio Explicit Formula an = a1 * r n-1 Ex: Write the explicit formula for each sequence First term: a1 = 7 Common ratio = 1/3 Explicit: an = a1 * r n-1 Now find the first five terms: a1 a2 a3 a4 a5 = = = = = 7(1/3) 7(1/3) 7(1/3) 7(1/3) 7(1/3) (1-1) (2-1) (3-1) (4-1) (5-1) = = = = = 7 7/3 7/9 7/27 7/81 Explicit Arithmetic Sequence Problem Find the 19th term in the sequence of 11,33,99,297 . . . an = a1 * r n-1 Common ratio = 3 Start with the explicit sequence formula Find the common ratio between the values. a19 = 11 (3) (19-1) a19 = 11(3)18 =4,261,626,379 Plug in known values Simplify Let’s try one Find the 10th term in the sequence of 1, -6, 36, -216 . . . an = a1 * r n-1 Start with the explicit sequence formula Common ratio = -6 a10 = 1 (-6) (10-1) a10 = 1(-6)9 = -10,077,696 Find the common ratio between the values. Plug in known values Simplify