9514 1404 393
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)
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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)