hello my name is zarinah and i am a student in the 9th year.(key stage 3)

i have a question about finding the nth term in fractions.

Here is the question

Each term in this sequence is made by increasing the numerator by 1 and the denominator by 3. here are the first five terms:

1/4, 2/7, 3/10, 4/13, 5/16, ...

(a) write an expression for the nth term:


i know what nth term means but don't know how to apply this to fractions.

please can you help!



I used a table to try to see what the pattern is. In the table below N is the numerator and D is the denominator.

  1st term 2nd term 3rd term 4th term 5th
N 1 1 + 1 1 +2(1) 1 + 3(1) 1 + 4(1)
D 4 4 + 3 4 + 2(3) 4 + 3(3) 4 + 4(3)

In the first term N = 1 and D = 3. In the second term you add 1 to N and 3 to D. (I wrote the numbers you add in red so that they would stand out.) In the third term you add another 1 to N and another 3 to D. Thus in the third term you have 1 plus 2 ones in the numerator and 4 plus 2 threes in the denominator. Likewise in the fourth term you have 1 plus 3 ones in the numerator and 4 plus 3 threes in the denominator.

I hope now you see the pattern. In every term the numerator is 1 + k(1) and the denominator is 4 + k(3) for some number k. The number k is one less than the term number, that is for the third term k = 2, for the fourth term k = 3, etc.

Now I can see the nth term, the numerator is 1 + (n-1)(1) and the denominator is 4 + (n-1)(3).

Write this fraction, simplify it and then check that when n = 1, 2, 3, ... you get the correct answers.