MIP1502
Assignment 3 2025
(Answer GUIDE) -
Due 18 July 2025
QUESTIONS WITH 100%
VERIFIED AND
CERTIFIED ANSWERS.
, 𝑴𝑰𝑷𝟏𝟓𝟎𝟐 𝑨𝒔𝒔𝒊𝒈𝒏𝒎𝒆𝒏𝒕 𝟑 𝟐𝟎𝟐𝟓 (𝑨𝒏𝒔𝒘𝒆𝒓 𝑮𝑼𝑰𝑫𝑬) − 𝑫𝒖𝒆 𝟏𝟖 𝑱𝒖𝒍𝒚 𝟐𝟎𝟐𝟓
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟏. 𝟏: 𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒄𝒐𝒏𝒔𝒆𝒄𝒖𝒕𝒊𝒗𝒆 𝒕𝒆𝒓𝒎𝒔
𝐿𝑒𝑡’𝑠 𝑒𝑥𝑡𝑟𝑎𝑐𝑡 𝑎𝑛𝑑 𝑎𝑛𝑎𝑙𝑦𝑧𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒:
𝑫𝒊𝒂𝒈𝒓𝒂𝒎 (𝒏) 𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔
1 1 1 0 0
2 4 3 1 0
3 9 5 3 1
4 16 7 6 3
5 25 9 10 6
6 36 11 ? 10
𝒂) 𝑭𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔:
𝐿𝑒𝑡’𝑠 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 𝑡ℎ𝑒 𝒇𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑎𝑡𝑡𝑒𝑟𝑛:
𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9, 36 − 25 = 11
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 3, 5, 7, 9, 11
⇒ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 2, 2, 2, 2 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 3 − 1 = 2, 5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2, 11 − 9 = 2
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 2, 2, 2, 2, 2 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1 − 0 = 1, 3 − 1 = 2, 6 − 3 = 3, 10 − 6 = 4
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1, 2, 3, 4
⇒ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1, 1, 1 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 0 − 0 = 0, 1 − 0 = 1, 3 − 1 = 2, 6 − 3 = 3, 10 − 6 = 4
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 0, 1, 2, 3, 4
⇒ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1, 1, 1, 1 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟏. 𝟐: 𝑪𝒍𝒂𝒔𝒔𝒊𝒇𝒚 𝒆𝒂𝒄𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏 𝒂𝒔 𝒍𝒊𝒏𝒆𝒂𝒓 𝒐𝒓 𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄
𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 (𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝐿𝑖𝑛𝑒𝑎𝑟 (𝑓𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟏. 𝟑: 𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒂𝒍 𝒋𝒖𝒔𝒕𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒄𝒍𝒂𝒔𝒔𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒐𝒇
𝑺𝒎𝒂𝒍𝒍 𝒂𝒏𝒅 𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔
, 𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝑇ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 1, 4, 9, 16, 25, 36 𝑖𝑠 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑜𝑓 𝒑𝒆𝒓𝒇𝒆𝒄𝒕 𝒔𝒒𝒖𝒂𝒓𝒆𝒔:
𝑇𝑛 = 𝑛2𝑇_𝑛 = 𝑛^2𝑇𝑛 = 𝑛2.
𝑇ℎ𝑒 𝒔𝒆𝒄𝒐𝒏𝒅 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒔 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝟐), 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑎 𝑘𝑒𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑜𝑓 𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆𝒔.
𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝑇ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 1, 3, 5, 7, 9, 11 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑏𝑦 𝒂 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒂𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝟐, 𝑠𝑜 𝑖𝑡 ℎ𝑎𝑠 𝑎 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝒇𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆.
𝑇ℎ𝑖𝑠 𝑖𝑠 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑜𝑓 𝒍𝒊𝒏𝒆𝒂𝒓 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆𝒔, 𝑤ℎ𝑒𝑟𝑒
𝑇𝑛 = 2𝑛 − 1𝑇_𝑛 = 2𝑛 − 1𝑇𝑛 = 2𝑛 − 1.
𝐻𝑒𝑛𝑐𝑒, 𝑠𝑚𝑎𝑙𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐, 𝑎𝑛𝑑 𝑏𝑙𝑎𝑐𝑘 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒
𝑏𝑒ℎ𝑎𝑣𝑖𝑜𝑟 𝑜𝑓 𝑡ℎ𝑒𝑖𝑟 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠.
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟐: 𝑹𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒊𝒏𝒈 𝑮𝒓𝒆𝒚 𝒂𝒏𝒅 𝑾𝒉𝒊𝒕𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆 𝑷𝒂𝒕𝒕𝒆𝒓𝒏𝒔
𝟏. 𝟏. 𝟐. 𝟏: 𝑫𝒆𝒔𝒄𝒓𝒊𝒃𝒆 𝒊𝒏 𝒘𝒐𝒓𝒅𝒔 𝒉𝒐𝒘 𝒆𝒂𝒄𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏 𝒈𝒓𝒐𝒘𝒔 𝒇𝒓𝒐𝒎 𝒐𝒏𝒆 𝒅𝒊𝒂𝒈𝒓𝒂𝒎 𝒕𝒐 𝒕𝒉𝒆 𝒏𝒆𝒙𝒕
𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
o 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐷𝑖𝑎𝑔𝑟𝑎𝑚 2, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑤𝑎𝑦:
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 2 → 1 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 3 → 3 (1 + 2)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 4 → 6 (3 + 3)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 5 → 10 (6 + 4)
o 𝑮𝒓𝒐𝒘𝒕𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏: 𝐸𝑎𝑐ℎ 𝑛𝑒𝑤 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑎𝑑𝑑𝑠 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑡ℎ𝑎𝑡 𝑖𝑠
o 𝒐𝒏𝒆 𝒎𝒐𝒓𝒆 𝑡ℎ𝑎𝑛 𝑡ℎ𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛.
𝑇ℎ𝑖𝑠 𝑖𝑠 𝑎 𝒕𝒓𝒊𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒏𝒖𝒎𝒃𝒆𝒓 𝒑𝒂𝒕𝒕𝒆𝒓𝒏 𝑤ℎ𝑒𝑟𝑒 𝑇𝑛 = 𝑛(𝑛 − 1)2𝑇_𝑛
= \𝑓𝑟𝑎𝑐{𝑛(𝑛 − 1)}{2}𝑇𝑛 = 2𝑛(𝑛 − 1) 𝑓𝑜𝑟 𝑛 ≥ 2𝑛 \𝑔𝑒𝑞 2𝑛 ≥ 2.
𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
o 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐷𝑖𝑎𝑔𝑟𝑎𝑚 3:
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 3 → 1 𝑤ℎ𝑖𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 4 → 3 (1 + 2)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 5 → 6 (3 + 3)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 6 → 10 (6 + 4)
o 𝑮𝒓𝒐𝒘𝒕𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏: 𝐿𝑖𝑘𝑒 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠, 𝑤ℎ𝑖𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑙𝑠𝑜 𝑓𝑜𝑙𝑙𝑜𝑤 𝑎 𝒕𝒓𝒊𝒂𝒏𝒈𝒖𝒍𝒂𝒓
o 𝒏𝒖𝒎𝒃𝒆𝒓 𝒑𝒂𝒕𝒕𝒆𝒓𝒏, 𝑏𝑢𝑡 𝑤𝑖𝑡ℎ 𝑎 𝑜𝑛𝑒 − 𝑠𝑡𝑒𝑝 𝑑𝑒𝑙𝑎𝑦.
𝑇ℎ𝑎𝑡 𝑖𝑠, 𝒘𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝒊𝒏 𝒅𝒊𝒂𝒈𝒓𝒂𝒎 𝒏𝒏𝒏 𝑒𝑞𝑢𝑎𝑙 𝒈𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝒊𝒏 𝒅𝒊𝒂𝒈𝒓𝒂𝒎 𝒏 − 𝟏𝒏 − 𝟏𝒏
− 𝟏.
Assignment 3 2025
(Answer GUIDE) -
Due 18 July 2025
QUESTIONS WITH 100%
VERIFIED AND
CERTIFIED ANSWERS.
, 𝑴𝑰𝑷𝟏𝟓𝟎𝟐 𝑨𝒔𝒔𝒊𝒈𝒏𝒎𝒆𝒏𝒕 𝟑 𝟐𝟎𝟐𝟓 (𝑨𝒏𝒔𝒘𝒆𝒓 𝑮𝑼𝑰𝑫𝑬) − 𝑫𝒖𝒆 𝟏𝟖 𝑱𝒖𝒍𝒚 𝟐𝟎𝟐𝟓
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟏. 𝟏: 𝑫𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒄𝒐𝒏𝒔𝒆𝒄𝒖𝒕𝒊𝒗𝒆 𝒕𝒆𝒓𝒎𝒔
𝐿𝑒𝑡’𝑠 𝑒𝑥𝑡𝑟𝑎𝑐𝑡 𝑎𝑛𝑑 𝑎𝑛𝑎𝑙𝑦𝑧𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒:
𝑫𝒊𝒂𝒈𝒓𝒂𝒎 (𝒏) 𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔
1 1 1 0 0
2 4 3 1 0
3 9 5 3 1
4 16 7 6 3
5 25 9 10 6
6 36 11 ? 10
𝒂) 𝑭𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔:
𝐿𝑒𝑡’𝑠 𝑐𝑜𝑚𝑝𝑢𝑡𝑒 𝑡ℎ𝑒 𝒇𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆𝒔 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑎𝑡𝑡𝑒𝑟𝑛:
𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9, 36 − 25 = 11
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 3, 5, 7, 9, 11
⇒ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 2, 2, 2, 2 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 3 − 1 = 2, 5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2, 11 − 9 = 2
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 2, 2, 2, 2, 2 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1 − 0 = 1, 3 − 1 = 2, 6 − 3 = 3, 10 − 6 = 4
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1, 2, 3, 4
⇒ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1, 1, 1 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 0 − 0 = 0, 1 − 0 = 1, 3 − 1 = 2, 6 − 3 = 3, 10 − 6 = 4
⇒ 𝐹𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 0, 1, 2, 3, 4
⇒ 𝑆𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠: 1, 1, 1, 1 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟏. 𝟐: 𝑪𝒍𝒂𝒔𝒔𝒊𝒇𝒚 𝒆𝒂𝒄𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏 𝒂𝒔 𝒍𝒊𝒏𝒆𝒂𝒓 𝒐𝒓 𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄
𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 (𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝐿𝑖𝑛𝑒𝑎𝑟 (𝑓𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔: 𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟏. 𝟑: 𝑴𝒂𝒕𝒉𝒆𝒎𝒂𝒕𝒊𝒄𝒂𝒍 𝒋𝒖𝒔𝒕𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒄𝒍𝒂𝒔𝒔𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒐𝒇
𝑺𝒎𝒂𝒍𝒍 𝒂𝒏𝒅 𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔
, 𝑺𝒎𝒂𝒍𝒍 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝑇ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 1, 4, 9, 16, 25, 36 𝑖𝑠 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑜𝑓 𝒑𝒆𝒓𝒇𝒆𝒄𝒕 𝒔𝒒𝒖𝒂𝒓𝒆𝒔:
𝑇𝑛 = 𝑛2𝑇_𝑛 = 𝑛^2𝑇𝑛 = 𝑛2.
𝑇ℎ𝑒 𝒔𝒆𝒄𝒐𝒏𝒅 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒔 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (𝟐), 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑎 𝑘𝑒𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑜𝑓 𝒒𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆𝒔.
𝑩𝒍𝒂𝒄𝒌 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
𝑇ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 1, 3, 5, 7, 9, 11 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑏𝑦 𝒂 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒂𝒎𝒐𝒖𝒏𝒕 𝒐𝒇 𝟐, 𝑠𝑜 𝑖𝑡 ℎ𝑎𝑠 𝑎 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝒇𝒊𝒓𝒔𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆.
𝑇ℎ𝑖𝑠 𝑖𝑠 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑜𝑓 𝒍𝒊𝒏𝒆𝒂𝒓 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆𝒔, 𝑤ℎ𝑒𝑟𝑒
𝑇𝑛 = 2𝑛 − 1𝑇_𝑛 = 2𝑛 − 1𝑇𝑛 = 2𝑛 − 1.
𝐻𝑒𝑛𝑐𝑒, 𝑠𝑚𝑎𝑙𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐, 𝑎𝑛𝑑 𝑏𝑙𝑎𝑐𝑘 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑙𝑖𝑛𝑒𝑎𝑟 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒
𝑏𝑒ℎ𝑎𝑣𝑖𝑜𝑟 𝑜𝑓 𝑡ℎ𝑒𝑖𝑟 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠.
𝑸𝒖𝒆𝒔𝒕𝒊𝒐𝒏 𝟏. 𝟏. 𝟐: 𝑹𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕𝒊𝒏𝒈 𝑮𝒓𝒆𝒚 𝒂𝒏𝒅 𝑾𝒉𝒊𝒕𝒆 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆 𝑷𝒂𝒕𝒕𝒆𝒓𝒏𝒔
𝟏. 𝟏. 𝟐. 𝟏: 𝑫𝒆𝒔𝒄𝒓𝒊𝒃𝒆 𝒊𝒏 𝒘𝒐𝒓𝒅𝒔 𝒉𝒐𝒘 𝒆𝒂𝒄𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏 𝒈𝒓𝒐𝒘𝒔 𝒇𝒓𝒐𝒎 𝒐𝒏𝒆 𝒅𝒊𝒂𝒈𝒓𝒂𝒎 𝒕𝒐 𝒕𝒉𝒆 𝒏𝒆𝒙𝒕
𝑮𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
o 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐷𝑖𝑎𝑔𝑟𝑎𝑚 2, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑤𝑎𝑦:
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 2 → 1 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 3 → 3 (1 + 2)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 4 → 6 (3 + 3)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 5 → 10 (6 + 4)
o 𝑮𝒓𝒐𝒘𝒕𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏: 𝐸𝑎𝑐ℎ 𝑛𝑒𝑤 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑎𝑑𝑑𝑠 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑡ℎ𝑎𝑡 𝑖𝑠
o 𝒐𝒏𝒆 𝒎𝒐𝒓𝒆 𝑡ℎ𝑎𝑛 𝑡ℎ𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛.
𝑇ℎ𝑖𝑠 𝑖𝑠 𝑎 𝒕𝒓𝒊𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒏𝒖𝒎𝒃𝒆𝒓 𝒑𝒂𝒕𝒕𝒆𝒓𝒏 𝑤ℎ𝑒𝑟𝑒 𝑇𝑛 = 𝑛(𝑛 − 1)2𝑇_𝑛
= \𝑓𝑟𝑎𝑐{𝑛(𝑛 − 1)}{2}𝑇𝑛 = 2𝑛(𝑛 − 1) 𝑓𝑜𝑟 𝑛 ≥ 2𝑛 \𝑔𝑒𝑞 2𝑛 ≥ 2.
𝑾𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔:
o 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑓𝑟𝑜𝑚 𝐷𝑖𝑎𝑔𝑟𝑎𝑚 3:
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 3 → 1 𝑤ℎ𝑖𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 4 → 3 (1 + 2)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 5 → 6 (3 + 3)
𝐷𝑖𝑎𝑔𝑟𝑎𝑚 6 → 10 (6 + 4)
o 𝑮𝒓𝒐𝒘𝒕𝒉 𝒑𝒂𝒕𝒕𝒆𝒓𝒏: 𝐿𝑖𝑘𝑒 𝑔𝑟𝑒𝑦 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠, 𝑤ℎ𝑖𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑙𝑠𝑜 𝑓𝑜𝑙𝑙𝑜𝑤 𝑎 𝒕𝒓𝒊𝒂𝒏𝒈𝒖𝒍𝒂𝒓
o 𝒏𝒖𝒎𝒃𝒆𝒓 𝒑𝒂𝒕𝒕𝒆𝒓𝒏, 𝑏𝑢𝑡 𝑤𝑖𝑡ℎ 𝑎 𝑜𝑛𝑒 − 𝑠𝑡𝑒𝑝 𝑑𝑒𝑙𝑎𝑦.
𝑇ℎ𝑎𝑡 𝑖𝑠, 𝒘𝒉𝒊𝒕𝒆 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝒊𝒏 𝒅𝒊𝒂𝒈𝒓𝒂𝒎 𝒏𝒏𝒏 𝑒𝑞𝑢𝑎𝑙 𝒈𝒓𝒆𝒚 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 𝒊𝒏 𝒅𝒊𝒂𝒈𝒓𝒂𝒎 𝒏 − 𝟏𝒏 − 𝟏𝒏
− 𝟏.
