There-exists-a-unique-positive-integer-a-for-which-The-sum-u-n-1-2023-n-2-na-5-is-an-integer-trictly-between-1000-amp-1000-find-a-u-

Question Number 193585 by York12 last updated on 16/Jun/23

T h e r e e x i s t s a u n i q u e p o s i t i v e i n t e g e r a f o r

w h i c h T h e s u m u = \underset{n = 1}{\sum^{2023}} ⌊ \frac{n^{2} - n a}{5} ⌋ i s a n i n t e g e r

t r i c t l y b e t w e e n - 1000 & 1000 f i n d a + u .

Answered by York12 last updated on 16/Jun/23

\boldsymbol u = \underset{\boldsymbol n = 1}{\sum^{2023}} ⌊ \frac{\boldsymbol n^{2} - \boldsymbol n a}{5} ⌋, ⌊ \frac{\boldsymbol n^{2} - \boldsymbol n a}{5} ⌋ = \frac{\boldsymbol n^{2} - \boldsymbol n a}{5} - {\frac{\boldsymbol n^{2} - \boldsymbol n a}{5}}

\boldsymbol w h e r e {⊛} \boldsymbol i n d i c a t e s \boldsymbol t h e \boldsymbol f r a c t i o n a l

\boldsymbol p a r t \boldsymbol f u n c t i o n

0 ⩽ {\frac{\boldsymbol n^{2} - \boldsymbol n a}{5}} < 1 \Rightarrow 0 ⩽ \underset{\boldsymbol n = 1}{\sum^{2023}} {\frac{\boldsymbol n^{2} - \boldsymbol n a}{5}} < 2023

- 1000 < \underset{\boldsymbol n = 1}{\sum^{2023}} ⌊ \frac{\boldsymbol n^{2} - \boldsymbol n a}{5} ⌋ < 1000 \Rightarrow 0 < \underset{\boldsymbol n = 1}{\sum^{2023}} (\frac{\boldsymbol n^{2} - \boldsymbol n a}{5}) - \underset{\boldsymbol n = 1}{\sum^{2023}} {\frac{\boldsymbol n^{2} - \boldsymbol n a}{5}} < 1000

\Rightarrow - 3023 < \underset{\boldsymbol n = 1}{\sum^{2023}} (\frac{\boldsymbol n^{2} - \boldsymbol n a}{5}) < 1000

\Rightarrow - 3023 < \frac{\underset{\boldsymbol n = 1}{\sum^{2023}} (\boldsymbol n^{2})}{5} - \frac{\boldsymbol a \underset{\boldsymbol n = 1}{\sum^{2023}} (\boldsymbol n)}{5} < 1000

\Rightarrow 5 \times (3023 + \frac{\underset{\boldsymbol n = 1}{\sum^{2023}} (\boldsymbol n^{2})}{5}) > \boldsymbol a > 5 \times (\frac{\underset{\boldsymbol n = 1}{\sum^{2023}} (\boldsymbol n)}{5} - 1000)

\Rightarrow 1349.007 > a > 1348.998 \Rightarrow a = 1349

∴ \boldsymbol u = \underset{\boldsymbol n = 1}{\sum^{2023}} ⌊ \frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5} ⌋ = \underset{\boldsymbol n = 1}{\sum^{2023}} (\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}) - \underset{\boldsymbol n = 1}{\sum^{2023}} {\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}}

\underset{\boldsymbol n = 1}{\sum^{2023}} (\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}) = 0 \Rightarrow \boldsymbol u = - \underset{\boldsymbol n = 1}{\sum^{2023}} {\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}}

\boldsymbol s o \boldsymbol n o w \boldsymbol w e \boldsymbol a r e \boldsymbol o n l y \boldsymbol i n t r e s t e d \boldsymbol i n \boldsymbol t h e \boldsymbol v a l u e

\boldsymbol o f \boldsymbol t h e \boldsymbol R e m a i n d e r \boldsymbol o f \boldsymbol t h e \boldsymbol e x p r e s s i o n

\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}, \boldsymbol o n e \boldsymbol o f \boldsymbol t h e \boldsymbol s p e c i a l \boldsymbol p r o p e r t i e s

\boldsymbol o f \boldsymbol m o d (5) & \boldsymbol m o d (10) \boldsymbol t h a t

\boldsymbol a_{\boldsymbol n} \boldsymbol a_{\boldsymbol n - 1} \boldsymbol a_{\boldsymbol n - 2} \boldsymbol a_{\boldsymbol n - 2} \dots \boldsymbol a_{1} \boldsymbol a_{0} \equiv \boldsymbol a_{0} \boldsymbol m o d (5)

\boldsymbol a_{\boldsymbol n} \boldsymbol a_{\boldsymbol n - 1} \boldsymbol a_{\boldsymbol n - 2} \boldsymbol a_{\boldsymbol n - 2} \dots \boldsymbol a_{1} \boldsymbol a_{0} \equiv \boldsymbol a_{0} \boldsymbol m o d (10)

\underset{\boldsymbol n = 1}{\sum^{2023}} {\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}} = ⌊ \frac{2023}{5} ⌋ \underset{\boldsymbol n = 1}{\sum^{5}} (\frac{(\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}) \boldsymbol m o d (5)}{5}) + \underset{\boldsymbol n = 1}{\sum^{2023 \boldsymbol m o d (5)}} (\frac{(\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}) \boldsymbol m o d (5)}{5})

\underset{\boldsymbol n = 1}{\sum^{5}} (\frac{(\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}) \boldsymbol m o d (5)}{5}) = 1, \underset{\boldsymbol n = 1}{\sum^{2023 \boldsymbol m o d (5)}} (\frac{(\frac{\boldsymbol n^{2} - 1349 \boldsymbol n}{5}) \boldsymbol m o d (5)}{5}) = 1

\Rightarrow \boldsymbol u = - 1 \times ((404 \times 1) + 1) = - 405

\Rightarrow \boldsymbol u + \boldsymbol a = 944

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