The sum of the first n term of a series is $3-[1/3^(n-1)]$ . How to obtain the expression for the $n$ th term of the series, Un?

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Answer 1 · 2018-01-11T10:49:22

$U_n=2/3^(n-1), or, 2*3^(1-n)$ .

Let, $S_n$ denote the sum of the first $n$ terms of the seq. ${U_n}$ .

Then, $S_n=[U_1+U_2+...+U_(n-1)]+U_n$ ,

$rArr S_n=S_(n-1)+U_n$ .

$:. U_n=S_n-S_(n-1)$ ,

$=[3-1/3^(n-1)]-[3-1/3^((n-1)-1)]$ ,

$=1/3^(n-2)-1/3^(n-1)$ ,

$=1/(3^n/3^2)-1/(3^n/3)$ ,

$=3^2/3^n-3/3^n=(9-3)/3^n$ ,

$=(2xx3)/3^n$ ,

$rArr U_n=2/3^(n-1), or, 2*3^(1-n)$ .

The sum of the first n term of a series is 3-[1/3^(n-1)]3-[1/3^(n-1)]. How to obtain the expression for the nnth term of the series, Un?