Question

How do I determine the sum of this sequence: 2; 6; 18;...; 3,188,646?

Answer 1

I set up the explicit formula as:
3188646=2*3^(n-1)

Answer 2

then I subtracted 2 to get
318644=3\[^{n-1}\]

Answer 3

\[3,188,644=3^{n-1}\]

Answer 4

I next did log of both sides and divides both by log3, but the left side was 13.6 and some more decimals, where did I mess up?

Answer 5

\[\frac{ \log_{3,188,644} }{ \log_{3} }=\frac{ (n-1) \log_{3} }{ \log_{3} }\]

Answer 6

Now what?

Answer 7

#1 There is insufficient information. We have no idea how many terms or what sort of sequence it is.

#2 If we are to ASSUME that it is a geometric sequence, THEN we can add it up.

2 = 2*3^0
6 = 2*3^1
18 = 2*3^2
...
3,188,646 = 2*3^13

Okay, that's a geometric series with 14 terms.

First Term: 2
Common Ratio: 3

Go!

Answer 8

Yes, sorry, it is a geometric sequence.
That way makes sense and thanks for showing it to me, but when I do the log way in my calculator I get:
13.63092918=n-1
Why isn't this a perfect 14?

Answer 9

Okay, the 14=n makes sense, so I'll proceed with that.
Now I do the geometric sum formula, no?

S=\[S = \frac{ 2 (1-3^{14})}{ 1-3 }\]

Answer 10

The ONLY thing for which we might need the logarithm is to find the 13.

You failed to remove the 2.

\(2\cdot3^{n} = 3,188,646\)

\(3^{n} = 1,594,323\)

\(n\cdot\log(3) = \log(1,594,323)\)

\(n = \dfrac{\log(1,594,323)}{\log(3)} = 6.2025763114/0.4771212547 = 13\)

Answer 11

n-1=13, right?

Answer 12

There are 14 terms in the sequence. The greatest exponent in the sequence is 13. This makes 14 the correct value in that formula.

Answer 13

\[S=\frac{ 2(1-4782969) }{ -2 }\]


Next, \[S=1-4782969\]


\[Final, S=4782969\]

Answer 14

Thanks a ton, tk. You've helped me before, so I knew you would be helpful.

Answer 15

errr, final =4782968

Answer 16

There are some sign problems in there, but you did manage to land in the right spot. Good work.