This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by 2 3 2 3 gives the next term. In other words, a n = a 1 ⋅ r n − 1 a n = a 1 ⋅ r n - 1 . Geometric Sequence: r = 2 3 r = 2 3 This is the form of a geometric sequence. a n = a 1 r n − 1 a n = a 1 r n - 1 Substitute in the values of a 1 = 1 2 a 1 = 1 2 and r = 2 3 r = 2 3 . a n = ( 1 2 ) ⋅ ( 2 3 ) n − 1 a n = ( 1 2 ) ⋅ ( 2 3 ) n - 1 Apply the product rule to 2 3 2 3 . a n = 1 2 ⋅ 2 n − 1 3 n − 1 a n = 1 2 ⋅ 2 n - 1 3 n - 1 Multiply 1 2 1 2 and 2 n − 1 3 n − 1 2 n - 1 3 n - 1 . a n = 2 n − 1 2 ⋅ 3 n − 1 a n = 2 n - 1 2 ⋅ 3 n - 1 Cancel the common factor of 2 n − 1 2 n - 1 and 2 2 . Tap for more steps... a n = 2 n − 2 3 n − 1 a n = 2 n - 2 3 n - 1 Substitute in the value of n n to find the n n th term. a 5 = 2 ( 5 ) − 2 3 ( 5 ) − 1 a 5 = 2 ( 5 ) - 2 3 ( 5 ) - 1 Simplify the numerator. Tap for more steps... a 5 = 8 3 ( 5 ) − 1 a 5 = 8 3 ( 5 ) - 1 Simplify the denominator. Tap for more steps... a 5 = 8 81 a 5 = 8 81
