To get the common ratio r, we divide any term by the previous one
One example: r = common ratio r = (second term)/(first term) r = (-2)/(1) r = -2
Another example: r = common ratio r = (third term)/(second term) r = (4)/(-2) r = -2 and we get the same common ratio every time
Side Note: each term is multiplied by -2 to get the next term
============================================================ Part B
Answer: The rule for the sequence is a(n) = (-2)^(n-1) where n starts at n = 1
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Explanation:
Recall that any geometric sequence has the nth term a(n) = a*(r)^(n-1) where the 'a' on the right side is the first term and r is the common ratio
The first term given to use is a = 1 and the common ratio found in part A above was r = -2 So, a(n) = a*(r)^(n-1) a(n) = 1*(-2)^(n-1) a(n) = (-2)^(n-1)
============================================================ Part C
Answer: The next three terms are 16, -32, 64
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Explanation:
We can simply multiply each previous term by -2 to get the next term. Do this three times to generate the next three terms
-8*(-2) = 16 16*(-2) = -32 -32*(-2) = 64
showing that the next three terms are 16, -32, and 64
An alternative is to use the formula found in part B
Plug in n = 5 to find the fifth term a(n) = (-2)^(n-1) a(5) = (-2)^(5-1) a(5) = (-2)^(4) a(5) = 16 .... which matches with what we got earlier
Then plug in n = 6 a(n) = (-2)^(n-1) a(6) = (-2)^(6-1) a(6) = (-2)^(5) a(6) = -32 .... which matches with what we got earlier
Then plug in n = 7 a(n) = (-2)^(n-1) a(7) = (-2)^(7-1) a(7) = (-2)^(6) a(7) = 64 .... which matches with what we got earlier
while the second method takes a bit more work, its handy for when you want to find terms beyond the given sequence (eg: the 28th term)
