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18 votes
In Tara's math class, each student made up a number pattern for classmates to identify. These are the numbers that Tara wrote. Find the pattern and identify the first and seventh numbers in the table.

a. first number:

b. seventh number:


Tara's Pattern


1st


2nd


3rd


4th


5th


6th


7th


3/8


5/16


9/32


17/64


33/128

User Lazka
by
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1 Answer

17 votes
17 votes

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Answer:

  • first number: 3/8
  • seventh number: 129/512
  • pattern: a[n] = (2^n +1)/(4×2^n)

Explanation:

The denominators have the sequence ...

8, 16, 32, 64, 128, ...

These values have a common ratio of 2, so this is an exponential pattern with a base of 2. Apparently, the multiplier is 4.

denominator = 4×2^n

The numerators have the sequence ...

3, 5, 9, 17, 33, ...

These values have no common ratio, but their differences are ...

2, 4, 8, 16

and those differences do have a common ratio of 2. That means an exponential value of 2^n will figure into the numerator value. When we compare the sequence 2^n = 2, 4, 8, 16, 32 to the actual numerators, we see the numerator values are ...

numerator = 2^n +1

So, the general term of Tara's fraction is ...

(2^n +1)/(4·2^n)

__

The first number is found in Tara's table: 3/8

The 7th number is (2^7+1)/(4×2^7) = 129/512

The pattern is ...

a[n] = (2^n +1)/(4×2^n)

User Amaris
by
2.8k points
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