Question

The sequence one sixth comma negative two sevenths comma three eighths comma negative four ninths and so on is given.

Answer 1

The next four terms in the sequence are 5/10, 6/11, 7/12, and 8/13. The explicit equation for f(n) is (n+5)/(n+11). The sign of f(53) is positive.

Step-by-step explanation:

The given sequence is 1/6, 2/7, 3/8, 4/9.

Part A: To find the next four terms in the sequence, we continue the pattern where the numerator increases by 1 and the denominator increases by 1 for each term. Therefore, the next four terms are 5/10, 6/11, 7/12, and 8/13.

Part B: The explicit equation for f(n) can be written as f(n) = (n+5)/(n+11).

Part C: To determine the sign of f(53) without calculating its value, we can observe that the numerator and denominator both increase as n increases. Therefore, the sign of f(53) is positive.

Answer 2

So,

Here we have the following sequence:

`(1)/(6),-(2)/(7),(3)/(8),-(4)/(9),\ldots`

We suppose that this pattern continues.

To list the following 4 terms in the sequence, we should notice that the numerator and denominator are increasing by 1 and the sign of each term changes between each one:

Notice that if the numerator is an odd number, the sign is positive. If the numerator is an even number, the sign is negative.

With this information, we can write the pattern:

So those are the 4 numbers that next in the sequence given.

The explicit equation for f(n) that represents the sequence is:

`f(n)=((-1)^(n+1)\cdot n)/(n+5)`

That's because the numerator increases 1 unit since the first term which is n=1, and the denominator increases 1 unit since it starts to count from 6, which is n+5 = (1+5 = 6) and so on.

The sign changes between each term, so (-1)^(n+1) represents that the sequence starts with a positive sign ( (-1)^(1+1) = (-1)^2 = 1 )) , and then alternates between each term.

And, if you notice, the sequence tells us that if the numerator is an odd number, the sign is positive. If the numerator is an even number, the sign is negative. So, as 53 is an odd number, the value of f(53) should be positive.