Question

How do you write the rule for the nth term given `-5,10,-15,20,...`?

Answer 1

`n^("th")" term "->a_n=5n(-1)^(n+1)`

Explanation:

Let the term count be `i`
Let the i'th term be `a_i`

Let the term the question askes for be `a_n`

Tip: Alternating positive an negative is achieved using something like: `(-1)^n`. So some variant on this will be included.

`color(brown)("Just considering the numbers. Will deal with the signs afterwards.")`

`a_1->5`
`a_2->10->10-5=5`
`a_3->15->15-10=5`
`a_4->20->20-15=5`

So this is an arithmetic sequence of type `a_i=5i`
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`color(brown)("Now we deal with the sign")`

`a_1>0 larr" "(-1)^2`
`a_2<0 larr" "(-1)^3`
`a_3>0 larr" "(-1)^4`

So for any `i" ; "a_i" includes "(-1)^(i+1)`

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`color(brown)("Putting it all together")`

For any `i"; "a_i=5i(-1)^(i+1)`

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`color(brown)("Putting it into the same format the question requires")`

`n^("th")" term "->a_n=5n(-1)^(n+1)`