Determine-1-a-r-ar-a-2r-ar-2-a-n-1-r-ar-n-1-

Question Number 5784 by Rasheed Soomro last updated on 27/May/16

D e t e r m i n e :

1 + \frac{a + r}{a r} + \frac{a + 2 r}{{a r}^{2}} + \dots + \frac{a + (n - 1) r}{{a r}^{n - 1}}

Answered by Yozzii last updated on 27/May/16

S (n) = 1 + \frac{a + r}{a r} + \frac{a + 2 r}{{a r}^{2}} + \frac{a + 3 r}{{a r}^{3}} + \dots + \frac{a + (n - 3) r}{{a r}^{n - 3}} + \frac{a + (n - 2) r}{{a r}^{n - 2}} + \frac{a + (n - 1) r}{{a r}^{n - 1}}

S (n) = 1 + (\frac{1}{r} + \frac{1}{a}) + (\frac{1}{r^{2}} + \frac{2}{a r}) + (\frac{1}{r^{3}} + \frac{3}{{a r}^{2}}) + (\frac{1}{r^{4}} + \frac{4}{{a r}^{3}}) + (\frac{1}{r^{5}} + \frac{5}{{a r}^{4}}) + \dots + (\frac{1}{r^{n - 3}} + \frac{n - 3}{{a r}^{n - 4}}) + (\frac{1}{r^{n - 2}} + \frac{n - 2}{{a r}^{n - 3}}) + (\frac{1}{r^{n - 1}} + \frac{n - 1}{{a r}^{n - 2}})

\Rightarrow r S (n) = r + (1 + \frac{r}{a}) + (\frac{1}{r} + \frac{2}{a}) + (\frac{1}{r^{2}} + \frac{3}{a r}) + (\frac{1}{r^{3}} + \frac{4}{{a r}^{2}}) + (\frac{1}{r^{4}} + \frac{5}{{a r}^{3}}) + \dots + (\frac{1}{r^{n - 4}} + \frac{n - 3}{{a r}^{n - 5}}) + (\frac{1}{r^{n - 3}} + \frac{n - 2}{{a r}^{n - 4}}) + (\frac{1}{r^{n - 2}} + \frac{n - 1}{{a r}^{n - 3}})

\Rightarrow (1 - r) S (n) = - r - \frac{2}{a} + 1 - 1 + \frac{1}{r} - \frac{1}{r} + \frac{1}{a} - \frac{r}{a} + \frac{1}{r^{2}} - \frac{1}{r^{2}} + \frac{2}{a r} - \frac{3}{a r} + \frac{1}{r^{3}} - \frac{1}{r^{3}} + \frac{3}{{a r}^{2}} - \frac{4}{{a r}^{2}} + \frac{1}{r^{4}} - \frac{1}{r^{4}} + \frac{4}{{a r}^{3}} - \frac{5}{{a r}^{3}} + \dots + \frac{1}{r^{n - 3}} - \frac{1}{r^{n - 3}} - \frac{1}{{a r}^{n - 4}} + 0 - \frac{1}{{a r}^{n - 3}} + \frac{1}{r^{n - 1}} + \frac{n - 1}{{a r}^{n - 2}}

(1 - r) S (n) = - r - \frac{1 + r}{a} - (\frac{1}{a r} + \frac{1}{{a r}^{2}} + \frac{1}{{a r}^{3}} + \dots + \frac{1}{{a r}^{n - 4}} + \frac{1}{{a r}^{n - 3}} + \frac{1}{{a r}^{n - 2}}) + \frac{1}{r^{n - 1}} + \frac{n}{{a r}^{n - 2}}

(1 - r) S (n) = - \frac{1 + r}{a} + \frac{1}{r^{n - 1}} + \frac{n}{{a r}^{n - 2}} - r - \frac{1}{a r} (1 + \frac{1}{r} + \frac{1}{r^{2}} + \dots \frac{1}{r^{n - 5}} + \frac{1}{r^{n - 4}} + \frac{1}{r^{n - 3}})

S (n) = \frac{1}{1 - r} (- \frac{1 + r}{a} + \frac{1}{r^{n - 1}} + \frac{n}{{a r}^{n - 2}} - r - \frac{1}{a r} (\frac{1 - \frac{1}{r^{n - 2}}}{1 - \frac{1}{r}}))

S (n) = \frac{1}{r - 1} (r + \frac{r + 1}{a} + \frac{1}{a r} (\frac{(r^{n - 2} - 1) / r^{n - 2}}{(r - 1) / r}) - \frac{1}{r^{n - 1}} - \frac{n}{{a r}^{n - 2}})

S (n) = \frac{1}{r - 1} (r + \frac{r + 1}{a} + \frac{r^{n - 2} - 1}{{a r}^{n - 2} (r - 1)} - \frac{1}{r^{n - 1}} - \frac{n}{{a r}^{n - 2}})

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E . g . n = 1

S (1) = \frac{1}{r - 1} (r + \frac{r + 1}{a} + \frac{r^{- 1} - 1}{{a r}^{- 1} (r - 1)} - 1 - \frac{1}{{a r}^{- 1}})

S (1) = \frac{1}{r - 1} (r - 1 + \frac{r + 1}{a} + \frac{r^{- 1} - 1}{a (1 - r^{- 1})} - \frac{r}{a})

S (1) = \frac{1}{r - 1} (r - 1 + \frac{1}{a} - \frac{1}{a})

S (1) = \frac{r - 1}{r - 1} = 1 (r \neq 1)

n = 2

S (2) = \frac{1}{r - 1} (r + \frac{r}{a} + \frac{1}{a} + 0 - \frac{1}{r} - \frac{2}{a})

S (2) = \frac{1}{r - 1} (\frac{r^{2} - 1}{r} + \frac{r - 1}{a})

S (2) = \frac{r + 1}{r} + \frac{1}{a} = 1 + \frac{1}{r} + \frac{1}{a} = 1 + \frac{a + r}{a r}

Commented by Rasheed Soomro last updated on 28/May/16

T_{HANK} S!