Question

50 points for whoever solves these!!!

Answer 1

Explanation:

Given:

Pattern:

`-(1)/(4), (2)/(5) , -(3)/(6), (4)/(7)`

Solution:

A) Your answer is correct. The pattern is where the top is increasing by 1 each time and the bottom is increasing by 1 as well but starting at 4 and it switches from negative to positive so the next numbers will be:

`-(5)/(8), (6)/(9) , -(7)/(10) , (8)/(11)`

B) Equation for f(n)

n is the number you are on.

So for the first number
`-(1)/(4)` n=1 because that is the first number in the pattern.

For
`(2)/(5)` , n=2 so on and so forth.

The top part of the equation is easier you can see that whatever n is the top is the same.

For the bottom part, add 3 because it's starts off 3 higher than n

So far we have:

`(n)/(n+3)`

Now we need to deal with the negative positive. you can't just put a - in front or multiply by a -1 because it would make it negative every time.

You would need to multiply by something that changes from neg to pos each time and we need to use exponents for that
`(-1)^(n)` See when n is odd, 1, 3, 5 it will produce a negative, but when n is even it will be positive.

Put it together:

`f(n) = (-1)^(n)((n)/(n+3) )`

C) As discussed at the end of B F(59) = negative because 59 is odd.

(-1)⁵⁹ will produce a negative answer so F(59 is negative

Answer 2

Part A: Assuming the pattern continues, the next four terms in the sequence can be calculated as follows:

5/8, -6/9, 7/10, -8/11

To find the pattern, we observe that the numerator starts with 1 and increases by 1 each time, while the denominator starts with 4 and increases by 1 each time. The sign alternates between positive and negative.

Part B: To write the explicit equation for f(n) to represent the sequence, we can express the numerator and denominator in terms of n.

The numerator follows the pattern: n, -(n+1), (n+2), -(n+3), ...

The denominator follows the pattern: n+3, (n+4), -(n+5), (n+6), ...

Therefore, the explicit equation for f(n) is:

f(n) = (-1)^(n+1) * (n / (n+3))

Part C: To determine the sign of 7^(59) without calculating its value, we can consider the parity of the exponent (59).

Since 59 is an odd number, (-1)^(59) will be negative. This is because any odd exponent of -1 will result in a negative value.

Therefore, the sign of 7^(59) is negative.

I HAVE ACCOMPLISHED 50 POINTS!!