Interesting Sequence

Algebra Level 2

Consider the following sequence $1, 1, \frac{1}{2}, 1, \frac{2}{3}, \frac{1}{3}, 1, \frac{3}{4}, \frac{1}{2}, \frac{1}{4}, \cdots,$ and let $A_n$ denote the $n^{\text{th}}$ term. What is the term $A_{164}?$

\frac{7}{10}

\frac{5}{8}

\frac{6}{11}

\frac{4}{9}

3 solutions

Yodji Fufuri
Apr 18, 2014

solution.. we can make the sequence like

1/1 , 2/2 , 1/2 , 3/3 , 2/3 , 1/3 , 4/4 , 3/4 , 2/4 , 1/4 .....

so, we will get arithmatic progression 1, 2, 3, 4 , 5 .....

term 164 = sum of arithmatic

164 = N/2 (2 first term + (n-1) D)

164 = N/2 (2 + (n-1))

328 = N^2 - N

N^2 - N - 328 = 0

make 328 = 342 because the end of term must be have the solution... it's kinda like 0.8 = 1

we do this to get real solution of the sequence..

so N = -19 and 18

because there is no way, total term is a minus, we take N = 18

term 1 = X/1

term 4 = X/4 (we put X because we don't know which number in term 4 we will take)

so that will make term 18 = X / 18

the answer is x/18 or (x/2)/(18/2)

(X/2) / 9

the answer must contain multiplied of 3 such as 18, 6, 9, 15 or else...

sorry, i doesn't have abilty to be a teacher so, i am sorry if my solution make you all confuse..

u r correct....better dont be a teacher....but u can be a mathematician

Max B - 7 years, 1 month ago

thanks dude,, but i want to a engineer,, :) it's allready my dream to make a machine that emitted oxygen, so maybe i can save the future of this earth,, :)

Yodji Fufuri - 7 years, 1 month ago

Already exists, scientists have found ways to "synthesize" photosynthesis and create "chips" that when exposed to sunlight, water and carbon dioxide produce oxygen from one side and hydrogen from the other .

Here's the link: http://en.wikipedia.org/wiki/Artificial_Photosynthesis

PS. This is not me trying to spoil your dreams by telling it exists, but me trying to help you develop your dream and this experiment further until we can create a fully functioning, affordable, air-producing machine. Any offense is not intended.

James Hadley - 7 years ago

Don't understand ur process yodji fufuri

subhankar das - 6 years, 11 months ago

Avi Aryan
May 29, 2014

We can see the sequence as - 1/1 --- 2/2 , 1/2 --- 3/3 , 2/3 , 1/3 --- 4/4 , 3/4 , 2/4 , 1/4 --- 5/5 , 4/5 .....

We can take them as 1 series , 2 series , 3 series ..... so on where (1 series) has 1 term , (2 series) has 2 terms, (3 series) has 3 terms and so on. For term no 164 , from simple hit and trial we see the nearest series completion ... sum till 17 = 17*(17+1)/2 = 153

So term 153 will be 1/17 , term 154 will be 18/18 and term 155 will be 17/18 . .... As term 155 has 17 as numerator, numerator of term 164 = 17 - (164-155) = 8

Term 164 = 8/18 = 4/9

Very simple and nice logic Avi Aryan well done

Avi Vijay - 6 years, 10 months ago

Amulu Kutima
Feb 27, 2014

its sooo very use full too study chlid

Please submit a proper solution.

Shabarish Ch - 7 years, 2 months ago

Interesting Sequence

3 solutions

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