SOLUTION: A tortoise travelled 4096 m in week one. Each week, it travelled half as far as the previous week. In what week will the tortoise travel 4 m? Please show me using a formula or equa

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Question 696834: A tortoise travelled 4096 m in week one. Each week, it travelled half as far as the previous week. In what week will the tortoise travel 4 m? Please show me using a formula or equation. I know the answer because I used the calculator but I still want to know how to find the answer using a formula or equation.I'm in grade 7 can you explain, because the previous time i didnt understand. Help!
Found 3 solutions by stanbon, josmiceli, MathTherapy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
A tortoise travelled 4096 m in week one. Each week, it travelled half as far as the previous week. In what week will the tortoise travel 4 m? Please show me using a formula or equation. I know the answer because I used the calculator but I still want to know how to find the answer using a formula or equation.
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Look at the pattern:
#1 term: 4096
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#2 term: 4096(1/2) = 2048
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#3 term: 4096(1/2)^2 = 1024
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....
....
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nth term: 4096(1/2)^(n-1)
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You want to find "n" when you are told the term is 4
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Solve: 4096(1/2)^(n-1) = 4
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Note: 4096 = 2*2*2*2*2...*2*2*2 = 2^12
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So: (2^(12))(1/2)^(n-2) = 4
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Then: (2^12))(2)^(2-n) = 4
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2^(14-n) = 2^2
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14-n = 2
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n = 12
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4 is the 12th term of the sequence.
================
Cheers,
Stan H.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Just to get a handle on this, i'll start
by saying
+W%5Bn%5D+=+a+%2F+%282%5En%29+
where +W%5Bn%5D+ is the distance traveled for week number +n+
I'll say that +n+ starts at +0+, and
adds 1 more for each week, +W+
+W%5B0%5D+=+a+%2F+%28+2%5E0+%29+
+W%5B1%5D+=+a+%2F+%28+2%5E1+%29+
+W%5B2%5D+=+a+%2F+%28+2%5E3+%29+
etc.
Notice that for any particulart week, +W%5Bn%5D+
The value is half that of the previous week
Now let +a+=+4096+
I want the condition:
+W%5Bn%5D+=+4096+%2F+%28+2%5En+%29+
+4+=+4096+%2F+%28+2%5En+%29+
+2%5En+=+4096+%2F+4+
+2%5En+=+1024+
You can use a calculator depending on what
functions it has. At the most basic, you can take
log base 10 of both sides
+log%28+2%5En+%29+=+log%28+1024+%29+
+n+=+log%28+1024+%29+%2F+log%282%29+
+n+=+3.0103+%2F+.30103+
+n+=+10+
------------
Notice that I said that week one started with +n+=+0+
so, I must add +1+ to each week number, so
in week number 11, the tortoise travels 4 m





Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

A tortoise travelled 4096 m in week one. Each week, it travelled half as far as the previous week. In what week will the tortoise travel 4 m? Please show me using a formula or equation. I know the answer because I used the calculator but I still want to know how to find the answer using a formula or equation.I'm in grade 7 can you explain, because the previous time i didnt understand. Help!

The terms represent a geometric sequence with the first term, or a%5B1%5D: 4,096, r, or common ratio: 1%2F2, and n: the week-number that the tortoise will travel 4 m

Geometric sequence: a%5Bn%5D+=+a%5B1%5Dr%5E%28n+-+1%29

Therefore, a%5Bn%5D+=+a%5B1%5Dr%5E%28n+-+1%29 becomes: a%5Bn%5D+=+4096%281%2F2%29%5E%28n+-+1%29

4+=+4096%281%2F2%29%5E%28n+-+1%29

4%2F4096+=+%281%2F2%29%5E%28n+-+1%29

1%2F1024+=+%281%2F2%29%5E%28n+-+1%29

log+%281%2F2%2C+%281%2F1024%29%29+=+n+-+1

log+%281%2F1024%29%2Flog%281%2F2%29+=+n+-+1

log+%281%2F1024%29%2Flog%281%2F2%29+=+10, so:

10 = n - 1

n = 10 + 1

The tortoise will travel 4 m in week n, or week highlight_green%2811%29

You can USE MS Excel to check!!

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