What is the next term in the geometric sequence 16, \(-4\), 1, \(-\frac{1}{4}\), … ?
Answer (2)
a 1 = 16 ; a 2 = − 4 ; a 3 = 1 ; a 4 = − 4 1 q = a 2 : a 1 q = − 4 : 16 = − 4 1 a n = a 1 ⋅ q n − 1 a n = 16 ⋅ ( − 4 1 ) n − 1 = 16 ⋅ ( − 4 1 ) n ⋅ ( − 4 1 ) − 1 = 16 ⋅ ( − 4 1 ) n ⋅ ( − 4 ) = − 64 ⋅ ( − 4 1 ) n = ( − 4 ) 3 ⋅ ( − 4 1 ) n = ( − 4 1 ) − 3 ⋅ ( − 4 1 ) n = ( − 4 1 ) n − 3
a 5 = ( − 4 1 ) 5 − 3 = ( − 4 1 ) 2 = 16 1 a 6 = ( − 4 1 ) 6 − 3 = ( − 4 1 ) 3 = − 64 1 ⋮
The next term in the geometric sequence 16, -4, 1, -1/4, ... is 16 1 . This is determined by identifying the common ratio as − 4 1 and applying it to the last term. Therefore, a 5 = − 4 1 × − 4 1 = 16 1 .
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