Progressoes Aritmeticas

❙❡q✉ê♥❝✐❛s

▼❆❚❊▼➪❚■❈❆ ✹
❈❖❆▲■ ✶✵✽✸
❋❡r♥❛♥❞♦ ❙✐❧✈❡✐r❛ ❆❧✈❡s1
1 ❈❆▼P❯❙ ❈❖❳■▼
■♥st✐t✉t♦ ❋❡❞❡r❛❧ ❞♦ ▼❛t♦ ●r♦ss♦ ❞♦ ❙✉❧

❈♦①✐♠✱ ✶✸ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✶✹

✶ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

❖✉t❧✐♥❡

✶

❙❡q✉ê♥❝✐❛s
■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦
P♦r ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛
❊①♣r❡ss❛♥❞♦ ❝❛❞❛ t❡r♠♦ ❡♠ ❢✉♥çã♦ ❞❡ s✉❛ ♣♦s✐çã♦
P♦r ♣r♦♣r✐❡❞❛❞❡ ❞♦s t❡r♠♦s

✷ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦

❖✉t❧✐♥❡

✶

❙❡q✉ê♥❝✐❛s
■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦
P♦r ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛
❊①♣r❡ss❛♥❞♦ ❝❛❞❛ t❡r♠♦ ❡♠ ❢✉♥çã♦ ❞❡ s✉❛ ♣♦s✐çã♦
P♦r ♣r♦♣r✐❡❞❛❞❡ ❞♦s t❡r♠♦s

✸ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦

❉✉❛s ❛♣❧✐❝❛çõ❡s f ❡ g sã♦ ✐❣✉❛✐s q✉❛♥❞♦ tê♠ ❞♦♠í♥✐♦s
✐❣✉❛✐s ❡ f (x) = g (x) ♣❛r❛ t♦❞♦ x ❞♦ ❞♦♠í♥✐♦✳ ❆ss✐♠✱
❞✉❛s s❡q✉ê♥❝✐❛s ✐♥✜♥✐t❛s f = (ai )i ∈ N∗ ❡ g = (bi )i ∈
N∗ sã♦ ✐❣✉❛✐s q✉❛♥❞♦ f (i) = g (i)✱ ✐st♦ é✱ ai = bi ♣❛r❛ t♦❞♦ i ∈ N∗ ✳ ❊♠ sí♠❜♦❧♦s✿ f = g ⇔ ai = bi , ∀i ∈ N∗

✹ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦

❖✉t❧✐♥❡

✶

❙❡q✉ê♥❝✐❛s
■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦
P♦r ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛
❊①♣r❡ss❛♥❞♦ ❝❛❞❛ t❡r♠♦ ❡♠ ❢✉♥çã♦ ❞❡ s✉❛ ♣♦s✐çã♦
P♦r ♣r♦♣r✐❡❞❛❞❡ ❞♦s t❡r♠♦s

✺ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦

■♥t❡r❡ss❛♠ à ▼❛t❡♠át✐❝❛ ❛s s❡q✉ê♥❝✐❛s ❡♠ q✉❡ ♦s t❡r♠♦s s❡ s✉❝❡❞❡♠ ♦❜❡❞❡❝❡♥❞♦ ❛ ❝❡rt❛ r❡❣❛✱ ✐st♦ é✱ ❛q✉❡❧❛s q✉❡ tê♠ ✉♠❛
❧❡✐ ❞❡ ❢♦r♠❛çã♦✳ ❊st❛ ♣♦❞❡ s❡r ❛♣r❡s❡♥t❛❞❛ ❞❡ três ♠❛♥❡✐r❛s✿
✶
✷
✸

P♦r ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛❀
❊①♣r❡ss❛♥❞♦ ❝❛❞❛ t❡r♠♦ ❡♠ ❢✉♥çã♦ ❞❡ s✉❛ ♣♦s✐çã♦❀
P♦r ♣r♦♣r✐❡❞❛❞ ❞♦s t❡♠♦s✳

✻ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦

❙ã♦ ❞❛❞❛s ❞✉❛s r❡❣r❛s✿ ✉♠❛ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♦ ♣r✐♠❡✐r♦ t❡r♠♦ (a1 ) ❡
♦✉tr❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❝❛❞❛ t❡r♠♦ (an ) ❛ ♣❛rt✐r ❞♦ ❛♥t❡❝❡ss♦r (an−1 )✳

✼ ✴ ✶✷

❙❡q✉ê♥❝✐❛s

■❣✉❛❧❞❛❞❡
▲❡✐ ❞❡ ❢♦r♠❛çã♦

❙ã♦ ❞❛❞❛s ❞✉❛s r❡❣r❛s✿ ✉♠❛ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♦ ♣r✐♠❡✐r♦ t❡r♠♦ (a1 ) ❡
♦✉tr❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❝❛❞❛ t❡r♠♦ (an ) ❛ ♣❛rt✐r ❞♦ ❛♥t❡❝❡ss♦r (an−1 )✳
❊①❡♠♣❧♦
❊s❝r❡✈❡r ❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛ f ❝✉❥♦s t❡r♠♦s ♦❜❡❞❡❝❡♠ à s❡❣✉✐♥t❡
❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛✿ a1 = 2 ❡ an = an−1 + 3✱ ∀n ∈ {2, 3, 4,