what is a possible explicit rule for the nth term of the sequence: 1/3, 1/2, 3/5, 2/3, 5/7?

This one is a bit tricky, but a really important trick with fractions is that they can be written in different ways. For example, 1/2 = 2/4. This is the trick that this question wants you to see:

1/3, 1/2, 3/5, 2/3, 5/7
is the same as
1/3, 2/4, 3/5, 4/6, 5/7

It is then much easier to see that a possible rule is n/(n+2) since the denominator in this pattern is always 2 more than the numerator.

Answer Link

MrRoyal MrRoyal · Answer 2 · 2022-04-15T14:33:26+02:00

The explicit rule for the nth term of the sequence is Tn = n/(n+2)

How to determine the explicit rule?

The sequence is given as:

1/3, 1/2, 3/5, 2/3, 5/7

Rewrite as:

1/3, 2/4, 3/5, 4/6, 5/7

Express the denominator as the sum of 2 and the numerator

1/(2 + 1), 2/(2 + 2), 3/(2 + 3), 4/(2 + 4), 5/(2 + 5)

Replace the occurrence with n

n/(n + 1), n/(n + 2), n/(n + 2), n/(n + 2), n/(n + 2)

The above means that the explicit rule of the sequence is:

Tn = n/(n+2)

Hence, the explicit rule for the nth term of the sequence is Tn = n/(n+2)