1-x-2023-1-x-x-2-2022-a-0-a-1-x-a-2-x-2-a-6067-x-6067-also-if-the-equations-a-n-1-mod-5-a-n-2-mod-5-n-0-1-2-6067-has-u-and-v-solutions-respectivley-then-prove-that-

Question Number 140606 by mathsuji last updated on 10/May/21

{(1 + x)}^{2023} {(1 - x + x^{2})}^{2022} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{6067} x^{6067}

a l s o i f t h e e q u a t i o n s

a_{n} \equiv 1 (m o d 5), a_{n} \equiv 2 (m o d 5), n = 0; 1; 2; \dots; 6067

h a s u a n d v s o l u t i o n s r e s p e c t i v l e y,

t h e n p r o v e t h a t \dots

\sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + \dots}}}} = \sqrt{1 + u \cdot v}

Commented by mathsuji last updated on 10/May/21

S i r m r . W p l i z \dots

Commented by mr W last updated on 10/May/21

\sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + \dots}}}} = 3

a_{n} = C_{k}^{2022} w i t h n = 3 k, k = 0, 1, . ., 2022

a_{n} = C_{k}^{2022} w i t h n = 3 k + 1

w e h a v e 4 t i m e s a_{n} w i t h a_{n} \equiv 1 m o d 5,

i . e . 2 t i m e s C_{0}^{2022} a n d 2 t i m e s C_{2}^{2022} .

w e h a v e 2 t i m e s a_{n} w i t h a_{n} \equiv 2 m o d 5,

i . e . 2 t i m e s C_{1}^{2022} .

t h a t m e a n s u = 4, v = 2

\sqrt{1 + u v} = \sqrt{1 + 4 \times 2} = 3

Commented by mathsuji last updated on 10/May/21

p e r f e c t S i r t h a n k s