Write the rule for the nth term given #2,-4,8,-10,14,...

Jimena Hatfield

Jimena Hatfield

Answered question

2022-09-11

Write the rule for the nth term given #2,-4,8,-10,14,...

Answer & Explanation

yamalwg

yamalwg

Beginner2022-09-12Added 17 answers

Given:
2,−4,8,−10,14
Look at the sequence of differences between successive terms:
−6,12,−18,24
The absolute values of these are in arithmetic progression with common difference 6, but with alternating signs.
Hence we can express our original sequence by a recursive rule:
{ a 1 = 2 a n + 1 = a n + ( - 1 ) n 6 n
How about a general formula for a n in terms of n?
Looking at the alternate terms we can observe that we have two interwoven arithmetic sequences - one (2,8,14,...) with a common difference of 6 and the other (−4,−10,−16,...) with a common difference of −6.
Hence we can write:
a n = { 6 n - 4 if n is odd 8 - 6 n if n is even
Note that if we subtract 2 from each of the expressions for odd and even n then we can make the expressions look a little more similar:
a n = { 2 - ( 6 - 6 n ) if n is odd 2 + ( 6 - 6 n ) if n is even
and hence:
a n = 2 + ( - 1 ) n 6 ( 1 - n )

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?