How can I solve this linear equality
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In 1-4, to determine whether a sequence is either arithmetic or geometric, you need to look at differences of consecutive terms (arithmetic) and ratios of consecutive terms (geometric). If you can't find it, the sequence will fall under the "neither" category.
For example, the differences between consecutive terms in the first sequence are
If the sequence was arithmetic, the difference between consecutive terms would have been the same constant throughout this list. But that's not the case, so this sequence is not arithmetic.
The ratios between consecutive terms are
The sequence would have been geometric if the list contained the same value throughout, but it doesn't. So this sequence is neither arithmetic nor geometric.
Meanwhile, in the second sequence, the differences are
so this sequence is arithmetic.
In 5-6, you know the sequences are arithmetic, so you know that they follow the recursive rule
For example, in the fifth sequence we know the first term is . The common difference between terms is
. So using the rule above, we have the pattern
and so on, so that the -th term is determined entirely by
with the formula
This means the 21st term in the fifth sequence is
The process is simple: identify and
, plug them into the formula above, then evaluate it at whatever
you need to use.