Question

PLEASE HELP!!!!!

Determine whether the sequence converges or diverges. If it converges, give the limit.

108, -54, 27, -27/2

Answer 1

It is a geometric sequence. Every next term can be found by multiplying the current one with a certain number. That number is -54/108, or 27/-54, or etc.
Of course you can simplify that number. It is the constant factor r.
Geometric sequences are convergent if |r| < 1.

Answer 2

r would be the number you are multiplying by correct?

Answer 3

Yes, that is correct!

Answer 4

so it is divergent because r >1 ?

Answer 5

Think again: r = -54/108 = 27/-54 = -1/2...

Answer 6

oh i thought it was the number you were multiplying by..

Answer 7

That is right! Only, that number is -1/2.

Answer 8

108 * -1/2 = -54
-54 * -1/2 = 27

Answer 9

these are the answer choices i have

Converges; 216

Diverges

Converges; 540

Converges; 72

do you understand what i mean?

Answer 10

I understand.
Because |-1/2|=1/2 <1, we have convergence.
I think with limit they mean the limit of the series, not the sequence.
The limit of the sequence is 0.
The limit of the series can be found by the formula: \(L=\dfrac{a}{1-r}\).
In this formula, a is the first term, and r the constant factor, so

\(L=\dfrac{108}{1-(-\frac{1}{2})}=\dfrac{108}{\frac{3}{2}}=108 \cdot \frac{2}{3}=72\)

Answer 11

oh my gosh, the honestly makes so much sense! thank you so much. this really cleared it up for me. thank you thank you thank you!!! :)

Answer 12

YW!