Understanding Geometric Sequences: Explicit and Recursive Formulas, Lecture notes of Analytical Geometry and Calculus

Download Understanding Geometric Sequences: Explicit and Recursive Formulas and more Lecture notes Analytical Geometry and Calculus in PDF only on Docsity! Geometric Sequences Explicit and Recursive Formulas Recall that a geometric sequence has a pattern of multiplying by the same value over and over again. We can model that type of sequence in two ways. One is called an explicit formula and the other a recursive formula. First lets look at explicit formulas. Explicit formula of a Geometric Sequences The explicit form of a sequence is used to find the general term, or "nth" term, by plugging in the number of term we want to know. The explicit form of a geometric sequence is ( ) 1 1 − = n n rtt where n t is the general term, 1t is the first term of the sequence, r is the common ratio, and n is the number of term to plug in. For example: In the equation ( ) 1 26 − = n n t 6 is the first term of the sequence and 2 is the common ratio. We could write out the first several terms of this sequence and get 6, 12, 24, 48, ... Suppose we want to know what the 10th term of this sequence is without having to go through finding all of the middle terms. We can first plug in 10 for n in the formula. Then use order of operations to evaluate for the tenth term. ( ) ( ) ( ) ( ) 3072 5126 26 26 26 10 10 9 10 110 10 1 = = = = = − − t t t t t n n So the tenth term of this sequence is 3072. Likewise we can write an explicit formula for a geometric sequence by plugging in the first term and common ratio into ( ) 1 1 − = n n rtt . For example: In the sequence 2, 6, 18, 54, ... To write the explicit formula for this sequence we need to know the first term and the common ratio. We can see that the first term is 2. To find the common ratio divide any successive pair of terms such that the second in the pair is always the numerator and the first in the pair is the denominator. 3 2 6 6&2 ==r 3 6 18 18&6 ==r 3 18 54 54&18 ==r Now we know that the common ratio is 3. Using 21 =t and 3=r we get ( ) 1 32 − = n n t