# Next Term Of Strange Sequence

I tutored a 10th grader and I was asked this puzzle and I had spent nearly an hour with it and got “no where”. Any one can crack it? Please let me know. Thank you.

Question: Find the $$14$$ th term of the sequence: $$\frac{1}{2}, \frac{3}{7}, \frac{1}{3}, \frac{5}{19}, \frac{3}{14}, ....$$.

• Hmm... I don't think simply tweaking the numbers to generalize the expression will work here... How did you attempt to solve this? I can't find any relations between the terms to call this a sequence, or maybe I don't know something which is required to solve. May 21, 2019 at 3:29
• @SoumalyaPramanik Hint, make the numerators 2,3,4,5,6,... May 21, 2019 at 4:25

$$\displaystyle\frac24\ ,\ \frac37\ ,\ \frac4{12}\ ,\ \frac5{19}\ ,\ \frac6{28}\ ,...$$

differences between denominators are $$3,5,7,9,...$$
(and numerators are consecutive integers $$2,3,4,5,6,...$$ )

$$\displaystyle\frac24,\frac37,\frac4{12},\frac5{19},\frac6{28}, \\ \displaystyle\frac7{39},\frac8{52},\frac9{67},\frac{10}{84},\frac{11}{103},\\ \displaystyle\frac{12}{124},\frac{13}{147},\frac{14}{172},\frac{15}{199},\frac{16}{128}$$

elementary watson, $$\displaystyle a_{14}=\frac{15}{199}$$ .

More generally, $$\displaystyle a_n = \frac{n+1}{n^2+3}$$ .

Using the formula, $$\displaystyle a_{14}=\frac{14+1}{14^2+3}=\frac{14+1}{196+3}=\frac{15}{199}$$ .

• Thank you, Mirko. My algebra is rusty and should wear glass to see far... May 21, 2019 at 4:26
• Thank you, I am happy I was lucky ... May 21, 2019 at 4:27
• Excellent, this didn't cross my mind at all 😂 @Mirko May 21, 2019 at 15:39