a-n-is-a-natural-number-sequence-for-n-0-that-satisfy-recurrence-relation-a-m-n-a-m-n-m-n-1-1-2-a-2m-a-2n-for-m-n-nonnegative-integers-Find-a-2016-

Question Number 157795 by naka3546 last updated on 28/Oct/21

{a_{n}} i s a n a t u r a l n u m b e r s e q u e n c e f o r n ⩾ 0 t h a t s a t i s f y

r e c u r r e n c e r e l a t i o n a_{m + n} + a_{m - n} - m + n = 1 + \frac{1}{2} (a_{2 m} + a_{2 n}),

f o r \forall m, n n o n n e g a t i v e i n t e g e r s .

F i n d a_{2016} .

Answered by Rasheed.Sindhi last updated on 30/Oct/21

\underset{\overset{}{A}}{\underset{⏟}{a_{m + n} + a_{m - n} - m + n = 1 + \frac{1}{2} (a_{2 m} + a_{2 n})}}

∙ \begin{matrix} a_{1} = 3 (G i v e n) \end{matrix}

∙ m = n : a_{2 n} + a_{0} = 1 + \frac{1}{2} (2 a_{2 n}) \Rightarrow a_{0} = 1

\begin{matrix} a_{0} = 1 \end{matrix}

▸ n = 0 : 2 a_{m} - m = 1 + \frac{1}{2} (a_{2 m} + a_{0})

1 + \frac{1}{2} a_{2 m} + \frac{1}{2} = 2 a_{m} - m

\frac{1}{2} a_{2 m} = 2 a_{m} - m - \frac{3}{2}

a_{2 m} = 4 a_{m} - 2 m - 3

\begin{matrix} a_{2 n} = 4 a_{n} - 2 n - 3 \end{matrix}

a_{2 (2 n)} = 4 a_{2 n} - 2 n - 3 = 4 (4 a_{n} - 2 n - 3) - 2 n - 3

a_{4 n} = 16 a_{n} - 10 n - 15

\begin{matrix} a_{4 n} = 16 a_{n} - 10 n - 15 \end{matrix}

R e p l a c i n g n b y 4 n

a_{4 (4 n)} = 16 a_{4 n} - 10 n - 15

a_{16 n} = 16 (16 a_{n} - 10 n - 15) - 10 n - 15

= 256 a_{n} - 170 n - 255

A g a i n r e p l a c i n g n b y 2 n :

a_{16 (2 n)} = 256 a_{2 n} - 170 (2 n) - 255

a_{32 n} = 256 (4 a_{n} - 2 n - 3) - 340 n - 255

= 1024 a_{n} - 512 n - 768 - 340 n - 255

= 1024 a_{n} - 852 n - 1023

\begin{matrix} \begin{matrix} a_{32 n} = 1024 a_{n} - 852 n - 1023 \end{matrix} \dots A \end{matrix}

▸ A : (m = 2 n) : a_{3 n} + a_{n} - n = 1 + \frac{1}{2} (a_{4 n} + a_{2 n})

2 a_{3 n} + 2 a_{n} - 2 n = 2 + a_{4 n} + a_{2 n}

2 a_{3 n} + 2 a_{n} - 2 n - 2 = 4 a_{2 n} - 4 n - 3 + 4 a_{n} - 2 n - 3

= 4 (4 a_{n} - 2 n - 3) - 4 n - 3 + 4 a_{n} - 2 n - 3

= 16 a_{n} - 8 n - 12 - 4 n - 3 + 4 a_{n} - 2 n - 3

= 20 a_{n} - 14 n - 18

2 a_{3 n} = 18 a_{n} - 12 n - 16

a_{3 n} = 9 a_{n} - 6 n - 8

\begin{matrix} a_{3 n} = 9 a_{n} - 6 n - 8 \end{matrix}

R e p l a c i n g n b y 3 n :

a_{3 (3 n)} = 9 a_{3 n} - 6 (3 n) - 8

a_{9 n} = 9 (9 a_{n} - 6 n - 8) - 18 n - 8

= 81 a_{n} - 72 n - 80

\begin{matrix} \begin{matrix} a_{9 n} = 81 a_{n} - 72 n - 80 \end{matrix} \dots B \end{matrix}

▸ A : {\begin{cases} m \to 4 n \\ n \to 3 n \end{cases} \Rightarrow \underset{= 1 + \frac{1}{2} (a_{8 n} + a_{6 n})}{a_{7 n} + a_{n} - 7 n + 3 n}

▸ 2 a_{7 n} + 2 a_{n} - 8 n = 2 + a_{8 n} + a_{6 n}

▸ 2 a_{7 n} + 2 a_{n} - 8 n - 2 = a_{4 (2 n)} + a_{3 (2 n)}

= 16 a_{2 n} - 10 (2 n) - 15 + 9 a_{2 n} - 6 (2 n) - 8

= 25 a_{2 n} - 32 n - 23 = 25 (4 a_{n} - 2 n - 3) - 32 n - 23

= 100 a_{n} - 50 n - 75 - 32 n - 23

= 100 a_{n} - 82 n - 98

▸ 2 a_{7 n} + 2 a_{n} - 8 n - 2 = 100 a_{n} - 82 n - 98

2 a_{7 n} = 98 a_{n} - 74 n - 96

\begin{matrix} \begin{matrix} a_{7 n} = 49 a_{n} - 37 n - 48 \end{matrix} \dots C \end{matrix}

▸ R e p l a c i n g n b y 9 n i n A

a_{32 (9 n)} = 1024 a_{9 n} - 852 (9 n) - 1023

a_{288 n} = 1024 (81 a_{n} - 72 n - 80) - 7668 n - 1023

a_{288 n} = 82944 a_{n} - 81396 n - 82943

A g a i n r e p l a c i n g n b y 7 n

a_{288 (7 n)} = 82944 a_{7 n} - 81396 (7 n) - 82943

a_{2016 n} = 82944 (49 a_{n} - 37 n - 48) - 569772 n - 82943

N o w r e p l a c i n g n b y 1

a_{2016 n} = 82944 (49 a_{1} - 37 (1) - 48) - 569772 (1) - 82943

a_{2016 n} = 82944 (49 (3) - 37 - 48) - 569772 - 82943

= 82944 (62) - 652715

\begin{matrix} \begin{matrix} \begin{matrix} a_{2016} = 4489813 \end{matrix} \end{matrix} \end{matrix}

Commented by naka3546 last updated on 29/Oct/21

I^{'} m s o s o r r y, s i r . I f o r g o t t o w r i t e t h a t a_{1} = 3 .

Commented by Rasheed.Sindhi last updated on 30/Oct/21

n a k a s i r, t h e a n s w e r i s c o m p l e t e

n o w . p l e a s e v e r i f y i t . I f {i t}^{'} s n o t

c o r r e c t t h e n p l e a s e c h e c k m y

a n s w e r f o r m i s t a k e s . T h e a p p r o a c h

i s c e r t a i n l y r i g h t .

Commented by naka3546 last updated on 30/Oct/21

y e s, {i t}^{'} s c o r r e c t, s i r .

Commented by Rasheed.Sindhi last updated on 30/Oct/21

T han k Y ou!