Question

Write an explicit formula for the geometric sequence.

sequence: 25, 5, 1, 1/5

@mathmale Can you explain this?

Answer 1

@kirbykirby Can you help?

Answer 2

\[a_n=\frac{25}{5^n}\]

Answer 3

Can you further explain?

Answer 4

Nicole, please identify the first term of this sequence. Call it "a sub 1."\[a _{1}=?\]

Answer 5

\[a_n=\frac{25}{5^n}\] for n=0,1,2,3

Answer 6

Okay, so each next term is a division of 5?

Answer 7

yes

Answer 8

Now that you have the first term, how do you get the second? Hint: You multiply the first term by what? Each succeeding term is a fractional multiple of the previous term.

Answer 9

Nicole, please identify the first term of this sequence. Call it "a sub 1."

Answer 10

So whenever you have a sequence you find the common ration and put it under the first term?

Answer 11

First term = 25

Answer 12

I suppose you can also use this notation\[a_n=\left\{\frac{25}{5^n}\right\}_{n=0} ^3\]

Answer 13

Yes, if we're talking about geometric sequences. Yes, the first term is 25.
From that first term, we get the next term, 5, by multiplying 25 by what common ratio?

Answer 14

So whenever you have a sequence you find the common ration and put it under the first term? To find the explicit formula..

Answer 15

5 @ MM

Answer 16

It is a G.P.
first term=25
common ratio r=5/25=1/5
\[tn=ar ^{n-1}=25\left( \frac{ 1 }{ 5 } \right)^{n-1}\]

Answer 17

Nicole, at this point the key is to find the common ratio. 25 times 5 is 125, not 5, so 5 could not be the common ratio. Hint" the common ratio is a fraction.

Answer 18

Wait, I thought what Kirby said was the answer and what I said was how you got it..

Answer 19

Surjithayer (and kirbykirby) have done this nicely, showing that the initial term is 25 and that the common ratio is 1/5.

Answer 20

Okay

Answer 21

That ones done right?

Answer 22

Yeah I think mathmale is giving you a more general idea on how to find these formulae. The sequence given though looked trivial to me so I just write the formula. But yeah you are just dividing by 5 every time. But if you have a more complicated sequence, then you can follow mathmale's advice (particularly if there's an "added constant")

Answer 23

Let's try explaining that in different words:

"The first term is 25. The next term is 5. By what number (ratio or fraction) must we multiply the first term to obtain the second?

You could write 25x=5, then solve for x: x=5/25 = 1/5.

Again, the common ratio is 1/5.

Answer 24

Okay, I understand that part now

Answer 25

Both kirbykirby and Surjithayer show a starting value for n.

If we use kirbykirby's model, then \[a _{n}=25(1/5)^n,\]with n starting at 0.

Answer 26

Wait, um.. Find x.. 0.2?

Answer 27

Actually, n goes to infinity, since the problem does not ask you to "find the first 3 or 4 terms".

Note that 1/5 = 0.2; you could use either (but not both) in writing your formula for the nth term of the sequence.

Answer 28

Summarizing, if we start with the geom. sequence 25, 5, 1, 1/5, 1/25,

the first term, a, is 25; the common ratio, r, is 1/5, and n is merely a counter.

Please look at both kirbykirby's and Surjithayer's formulas for a-sub-n and then choose whichever is clearer to you.

Answer 29

Okay, thank you! @mathmale @kirbykirby
Could you help me with another?

Answer 30

Please post it, and, as before, I'll help you all I can. I'm helping a number of other students simultaneously. Nice work, Nicole.

Answer 31

Okay!