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

1 Answer
Feb 20, 2017

Answer:

#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#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#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)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Putting it all together")#

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

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

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