a In an arithmetic sequence there is a common difference separating consecutive terms. Since we have been given the first two terms, we can find this common difference and consequently the following two terms.
In this form, t(1) is the first term and m is the common difference. Examining the sequence, we can identify the first term as t(1)=4 and we know that the common difference is m=4. By substituting these values into the formula, we can write our equation.
t(n)=4+4(n−1)
b In a geometric sequence there is a common ratio separating consecutive terms. Since we have been given the first two terms we can find this common ratio and consequently the following two terms.
A geometric sequence in standard form is written in the following format.
t(n)=t(1)an−1
In this form t(1) is the first term and a is the common ratio. Examining the sequence, we can identify the first term as t(1)=4 and we know that the common ratio is m=2. By substituting these values into the formula we can write our equation.
t(n)=4(2)n−1
c We need to find a sequence which is neither arithmetic nor geometric, but still starts with 4,8,… Notice that if a sequence is arithmetic, its values will either only increase, only decrease, or stay the same (if the common difference is 0). We can avoid each case if our sequence switches between 4 and 8.
As we can see, this sequence has no common difference because the differences between consecutive terms alternate between 4 and -4. Remember that we also need to know that it has no common ratio. Let's check whether we already have a solution or if we need to adjust the sequence somehow.
Like in the case of differences we also have no common ratio, as ratios between consecutive terms alternate between 2 and 0.5. Therefore, we found an example of a sequence with no common ratio and no common difference 4, but which still starts with 4,8,…
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