Question

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

60, 10, 5/8 , 5/18 , ...

Converges; 72
Converges; 0
Diverges
Converges; -15540

2.) find an explicit rule for the nth term of the sequence.

1, 3, 9, 27, ...

an = 1 • 3n + 1
an = 3 • 1n - 1
an = 1 • 3n - 1
an = 1 • 3n

3. Find an explicit rule for the nth term of the sequence.
-5, -25, -125, -625,
an = -5 • 5n
an = 5 • -5n + 1
an = 5 • -5n
an = -5 • 5n - 1

Answer 1

Your first sequence
60, 10, 5/8 , 5/18 ,
does not have a common ratio, so it is not a simple geometric sequence
in other words to get from 60 to 10, multiply by 1/6
then from 10 multiply by 1/6 to get 10/6 = 5/3
but your third term is 5/8.

Answer 2

But each term is smaller the the one previous to it. So it is not diverging (growing larger and larger). So we can rule out choices about diverging
and it is definitely not approaching 72 or -15540

Answer 3

@phi Converges; 0??

Answer 4

If it continues getting smaller, (smaller and smaller fractions), it approaches 0

Answer 5

@phi what about number 3 and 2

Answer 6

For
2.) find an explicit rule for the nth term of the sequence. 1, 3, 9, 27, ...

I would look at the differences: 3-1= 2 9-3= 6 27-9 = 18
that is not a constant difference.

Then I would look at ratios: 3/1 = 3, 9/3 = 3; 27/9 = 3
that looks promising.
To get from one term to the next, what do you multiply by ?

Answer 7

an = 1 • 3n - 1 @phi

Answer 8

You should put in ^ like this: 3^(n-1)
other wise it looks like 3*n -1 which is different (and hard for me to figure out what you mean)

Answer 9

to figure out that rule I would like the number of the term, starting at 1
1 then the value

1 1
2 1*3 (I used the rule, multiply the previous term by 3)
3 1*3*3
4 1*3*3*3

if you know about exponents, when you multiply 3*3, you can write 3^2
and 3*3*3 is 3^3
so the table looks like this
1 1
2 1*3^1
3 1*3^2
4 1*3^3

now notice the exponent on the 3 is 1 less than the number of the term
1 1*3^(1-1)
2 1*3^(2-1)
3 1*3^(3-1)
4 1*3^(4-1)
n 1*3^(n-1) for term n, this is the value

Answer 10

^^ @phi thank u so much what did u get for 3

Answer 11

3. Find an explicit rule for the nth term of the sequence. -5, -25, -125, -625,

what do you multiply by to get the next term?

Answer 12

an = 5 • -5n + 1 @phi is it this one?

Answer 13

when n is 1 , using that rule what do you get?
\[ 5 \cdot -5^{n+1} \]
replace n with 1 and you get
\[ 5 \cdot -5^2 \]
which means
\[ 5 \cdot - (5\cdot 5) = - 5 \cdot 5 \cdot 5 = -125 \]
but we want -5 as the first term (when n=1)

Answer 14

a=-5
r=-25/-5=5
\[an=ar ^{n-1}\]

Answer 15

You should try to see how to do the problem
(or just memorize the formula surji posted... but that tends to go in one ear and out the other if you don't understand what is going on)