it is given that Σ_(r=1) ^n U_n = ((1+3^(2n+2) −2×5^(n+1) )/8) where U_n is the n^(th) term of a sequence find the simplified expression for U


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Question Number 33154 by Rio Mike last updated on 11/Apr/18

$i t i s g i v e n t h a t \underset{r = 1}{\sum^{n}} U_{n} = \frac{1 + 3^{2 n + 2} - 2 \times 5^{n + 1}}{8}$ $w h e r e U_{n} i s t h e n^{t h} t e r m o f a s e q u e n c e$ $f i n d t h e s i m p l i f i e d e x p r e s s i o n f o r U_{n}$
Commented by prof Abdo imad last updated on 11/Apr/18

$w e h a v e \sum_{k = 1}^{n} u_{k} = \frac{1 + 3^{2 n + 2} - 2 . 5^{n + 1}}{8} a l s o$ $\sum_{k = 1}^{n - 1} u_{k} = \frac{1 + 3^{2 n} - 2 . 5^{n}}{8} \Rightarrow$ $u_{1} + u_{2} + . . . . . + u_{n} - u_{1} - u_{2} - . . . - u_{n - 1}$ $= \frac{1 + 3^{2 n + 2} - 2 . 5^{n + 1} - 1 - 3^{2 n} + 2 . 5^{n}}{8} \Rightarrow$ $u_{n} = \frac{3^{2 n} (9 - 1) - 8 . 5^{n}}{8} = 9^{n} - 5^{n}$ $u_{n} = 9^{n} - 5^{n} .$
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