Find the next three terms of the given sequences and compute the common ratio with complete solutions.
1) 1/2, 2, 4, ____, ____, ____
2) 10, 5, ____, ____, ____
3) 7, 14, ____, ____, ____
Answer (1)
To solve these problems, we need to determine the next three terms of each sequence and compute the common ratio. Each sequence given is geometric, meaning each term is found by multiplying the previous term by a common ratio.
Sequence: 2 1 , 2 , 4 , …
Determine the common ratio: The common ratio r is found by dividing the second term by the first term: r = 2 1 2 = 2 × 1 2 = 4
Next terms:
4 × 4 = 16
16 × 4 = 64
64 × 4 = 256
Therefore, the next three terms are 16 , 64 , 256 .
Sequence: 10 , 5 , …
Determine the common ratio: The common ratio r is found by dividing the second term by the first term: r = 10 5 = 2 1
Next terms:
5 × 2 1 = 2.5
2.5 × 2 1 = 1.25
1.25 × 2 1 = 0.625
Thus, the next three terms are 2.5 , 1.25 , 0.625 .
Sequence: 7 , 14 , …
Determine the common ratio: The common ratio r is found by dividing the second term by the first term: r = 7 14 = 2
Next terms:
14 × 2 = 28
28 × 2 = 56
56 × 2 = 112
Therefore, the next three terms are 28 , 56 , 112 .
To summarize, the next three terms for each sequence are:
Sequence 1: 16 , 64 , 256
Sequence 2: 2.5 , 1.25 , 0.625
Sequence 3: 28 , 56 , 112
