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

9514 1404 393

Step-by-step 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)

it has the same slope and a different y-intercept

hope this : )

they used 313 gallons of paint