Define-the-sequence-a-n-by-the-recursive-formula-a-n-1-ca-n-nr-n-1-n-Z-n-1-with-a-1-h-and-c-r-h-C-c-r-0-Find-a-n-in-terms-of-n-

Question Number 3974 by Yozzii last updated on 26/Dec/15

D e f i n e t h e s e q u e n c e {a_{n}} b y t h e

r e c u r s i v e f o r m u l a

a_{n + 1} = {c a}_{n} - {n r}^{n - 1} (n \in Z, n ⩾ 1)

w i t h a_{1} = h a n d c, r, h \in C, c, r \neq 0 .

F i n d a_{n} i n t e r m s o f n .

Commented by Rasheed Soomro last updated on 26/Dec/15

a_{n + 1} = {c a}_{n} - {n r}^{n - 1} (n \in Z, n ⩾ 1)

n \to n - 1

a_{n} = {c a}_{n - 1} - (n - 1) r^{n - 2}

a_{2} = c h - 1

a_{3} = c (c h - 1) - 2 r = c^{2} h - c - 2 r

a_{4} = c (c^{2} h - c - 2 r) - 3 r^{2} = c^{3} h - c^{2} - 2 c r - 3 r^{2}

a_{5} = c (c^{3} h - c^{2} - 2 c r - 3 r^{2}) - 4 r^{3} = c^{4} h - c^{3} - 2 c^{2} r - 3 {c r}^{2} - 4 r^{3}

O b s e r v i n g / C o p y i n g p a t t e r n

{\begin{cases} a_{1} = h \\ \overset{n ⩾ 2}{a_{n}} = {- c^{n - 2} - 2 c^{n - 3} r - 3 c^{n - 4} r^{2} - \dots - k c^{n - k - 1} r^{k - 1}} + c^{n - 1} h, 1 ⩽ k ⩽ n \end{cases}

Commented by prakash jain last updated on 26/Dec/15

From {Rasheed}^{'} s contribution .

a_{n} = - c^{n - 2} - 2 c^{n - 3} r - 3 c^{n - 4} r^{2} - \dots - k c^{n - k - 1} r^{k - 1} + c^{n - 1} h

a_{n} = c^{n - 1} h - \underset{k = 0}{\sum^{n - 2}} (n - 2 - k) c^{k} r^{n - 2 - k}

\frac{r}{c} a_{n} = \frac{c^{n - 1} h}{r} - \underset{k = 0}{\sum^{n - 2}} (n - 2 - k) c^{k - 1} r^{n - 1 - k}

b_{n} = c^{n - 2} + 2 c^{n - 1} r + 3 c^{n - 2} r^{2} + 4 c^{n - 3} r^{3} + . . + (n - 1) r^{n - 2}

\frac{r}{c} b_{n} = + c^{n - 1} r + 2 c^{n - 2} r + 3 c^{n - 3} r^{3} + . . + (n - 2) r^{n - 2} + (n - 1) \frac{r^{n - 2}}{c}

b_{n} (1 - \frac{r}{c}) = c^{n - 2} + c^{n - 1} r + c^{n - 2} r^{2} + \dots + r^{n - 2} + (n - 1) \frac{r^{n - 2}}{c}

r e d i s a GP with Acommon ratio \frac{r}{c} t e r m (n - 1)

b_{n} (\frac{c - r}{c}) = \frac{c^{n - 2} (\frac{r^{n - 1}}{c^{n - 1}} - 1)}{\frac{r}{c} - 1} + (n - 1) \frac{r^{n - 2}}{c}

b_{n} (\frac{c - r}{c}) = \frac{(r^{n - 1} - c^{n - 1})}{r - c} + (n - 1) \frac{r^{n - 2}}{c}

b_{n} = - \frac{c (r^{n - 1} - c^{n - 1})}{{(c - r)}^{2}} + \frac{(n - 1) r^{n - 2}}{(c - r)}

a_{n} = c^{n - 1} h - b_{n}

a_{n} = c^{n - 1} h + \frac{c (r^{n - 1} - c^{n - 1})}{{(c - r)}^{2}} - \frac{(n - 1) r^{n - 2}}{(c - r)}

Commented by Yozzii last updated on 27/Dec/15

A w e s o m e .

Commented by Rasheed Soomro last updated on 27/Dec/15

G rea T!