# 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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To obtain the expression for the nth term of the series ( U_n ), subtract the sum of the first ( (n-1) ) terms from the sum of the first ( n ) terms, which gives:

[ U_n = \text{Sum of first n terms} - \text{Sum of first (n-1) terms} ]

Substituting the given expression for the sum of the first n terms:

[ U_n = \left(3 - \frac{1}{3^{(n-1)}}\right) - \left(3 - \frac{1}{3^{(n-1) - 1}}\right) ]

[ U_n = \left(3 - \frac{1}{3^{(n-1)}}\right) - \left(3 - 3^{(n-1)}\right) ]

[ U_n = \frac{1}{3^{(n-1)}} - 3^{(n-1)} ]

Therefore, the expression for the nth term of the series is:

[ U_n = \frac{1}{3^{(n-1)}} - 3^{(n-1)} ]

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