it-is-given-that-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-n-

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} + \dots . . + u_{n} - u_{1} - u_{2} - \dots - 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} .