Find a formula for the general term `T_(n)`  of the sequence. {2/3, 4/9, 8/27...}

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rakesh05's profile pic

rakesh05 | High School Teacher | (Level 1) Assistant Educator

Posted on

The terms of the given sequence are   `2/3,4/9,8/27, ......`

Here we can investigate the numerators and denominators separately.

The numerator sequence is   `2,4,8, ......`

i.e.   `2^1,2^2,2^3,......`

So the general term by this pattern is  =`2^n` .

Now the denominator sequence is    `3,9,27, ........`

   i.e.  `3^1,3^2,3^3,.....`

So the general term of this pattern is =`3^n` .

So the general term of the given sequence is  =`2^n/3^n`

                       or,            `(2/3)^n`

So,    `T_n=(2/3)^n` .

oldnick's profile pic

oldnick | (Level 1) Valedictorian

Posted on

Is showat once  the ratio   `T_(n+1)/T_n=2/3=cost`

so i'ts a geometrical sequence:

 

`T_(n+k)=(2/3)^k T_n`

Thus:

`T_n=(2/3)^(n-1)T_1=(2/3)^(n-1) (2/3)=(2/3)^n`

pramodpandey's profile pic

pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

Let sequence is denoted by `c_n`

`c_1=(2/3)^1`

`c_2=4/9=(2/3)^2`

`c_3=8/27=(2/3)^3`

`...`

`..`

`...`

`c_n=(2/3)^n`

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