Question

What is the most apparent nth term of this sequence (assume that n begins with 1):

1/2, -4/3, 9/4, -16/5

Answer 1

How do you want to start

Answer 2

I don't know where to start

Answer 3

Whats the pattern for the numerators

Answer 4

ignoring the negative for now

Answer 5

1+3=4 4+5=9 9+7=16

Answer 6

true, but whats 1^2, 2^2, 3^2, 4^2

Answer 7

1,4 ,9,16 so is n squared? is it negative n?

Answer 8

no? you alternate between + and - right?

Answer 9

the first part is correct i mean

Answer 10

try and solve the denom and then we'll look at the negative

Answer 11

in terms of n

Answer 12

n+1 for the denominator

Answer 13

good

Answer 14

so whats (-1) raised to an even power?

Answer 15

-1

Answer 16

(-1)*(-1) = (-1)^2 = (-1)^even

Answer 17

= 1 right?

Answer 18

(-1)(-1)(-1) = 1

is that what youre asking

Answer 19

when you multiply -1 times itself an even number of times it become 1, and if you do it an odd amount of times it stays -1 right?

Answer 20

yes

Answer 21

so you have +, -, +, -

Answer 22

if you take (-1)^n what pattern is it

Answer 23

it stays -1

Answer 24

calculate for n=1, n=2, n=...

Answer 25

for 1/2 n=1, -4/3 n=2, 9/4 n=3

Answer 26

wait so what would the formula be for the numerator?

Answer 27

n^2

Answer 28

look at trying to raise (-1) to something to produce the pattern +, -, +, -

Answer 29

you start with n=1, so what if you do (-1)^n

Answer 30

-1^1 = -1

Answer 31

but we know that -1 ^ even is positive, can you add something to n to make the exponent even in the first term?

Answer 32

\[(-1^{n+1})\frac{ n^2 }{ n+1 }, n \epsilon \mathbb{Z}^+\]