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Question Number 1789 by 112358 last updated on 26/Sep/15

I f ∣ r ∣\neq 1, f i n d a n e x p r e s s i o n f o r

T_{n} (r), w h e r e

T_{n} (r) = 1 + r^{2} + r^{3} + r^{4} + r^{6} + r^{8} + r^{9}

+ r^{10} + r^{12} + r^{14} + r^{15} + r^{16} +

\dots + r^{6 n} .

Answered by Rasheed Soomro last updated on 27/Sep/15

T_{n} (r) =

{1 + r^{2} + r^{4} + \dots + r^{6 n}} + {1 + r^{3} + r^{6} + \dots + r^{6 n}} - {1 + r^{6} + r^{12} + \dots + r^{6 n}}

F o r m u l a :

S = a + a r + {a r}^{2} + \dots + {l r}^{- 1} + l = \frac{r l - a}{r - 1}

= {\frac{r^{2} . r^{6 n} - 1}{r^{2} - 1}} + {\frac{r^{3} . r^{6 n} - 1}{r^{3} - 1}} - {\frac{r^{6} . r^{6 n} - 1}{r^{6} - 1}}

= \frac{r^{6 n + 2} - 1}{r^{2} - 1} + \frac{r^{6 n + 3} - 1}{r^{3} - 1} - \frac{r^{6 n + 6} - 1}{r^{6} - 1}