IOQM and Engineering Entrance - Progressions Part1

Problems in progressions - Practice for Olympiad and various exams
Mathematics Progression part - 1
for IOQM and Engineering Entrance.
For students from Class 8 to Class 12.
𝐴 𝑛𝑢𝑚𝑏𝑒𝑟 32 𝑖𝑠 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑖𝑛𝑡𝑜 4 𝑝𝑎𝑟𝑡𝑠 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑖𝑛 𝐴.𝑃.. 𝑇ℎ𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑎𝑛𝑑 𝑓𝑜𝑢𝑟𝑡ℎ 𝑡𝑜 𝑡ℎ𝑎𝑡 𝑜𝑓 𝑠𝑒𝑐𝑜𝑛𝑑 𝑎𝑛𝑑 𝑡ℎ𝑖𝑟𝑑 𝑖𝑠 7:15. 𝐿𝑜𝑤𝑒𝑠𝑡 𝑝𝑎𝑟𝑡 𝑖𝑠..?
[𝑱𝑬𝑬 𝑴𝑨𝑰𝑵] 𝐿𝑒𝑡 𝑎1, 𝑎2, 𝑎3,…….,𝑎49 𝑏𝑒 𝑖𝑛 𝐴.𝑃. 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∑_(𝑘=0)^12▒〖𝑎4𝑘+1〗=416 𝑎𝑛𝑑 𝑎9+𝑎43=66. 𝐼𝑓 𝑎12 + 𝑎22 + ……..+ 𝑎172 = 140 𝑚, 𝑡ℎ𝑒𝑛 𝑚 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜
𝐴 𝑠𝑒𝑟𝑖𝑒𝑠 𝑜𝑓 384 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 ℎ𝑎𝑠 𝑎 𝑠𝑢𝑚 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑓𝑜𝑢𝑟𝑡ℎ 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑢𝑚 𝑓𝑜𝑟 𝑡ℎ𝑖𝑠 𝑠𝑒𝑟𝑖𝑒𝑠.
𝐴𝑛 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝐺𝑃 𝑖𝑠 𝑓𝑜𝑟𝑚𝑒𝑑 𝑏𝑦 𝑡ℎ𝑟𝑒𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠. 𝑇ℎ𝑒 𝑛𝑒𝑤 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑓𝑜𝑟𝑚 𝑎𝑛 𝐴𝑃 𝑖𝑓 𝑡ℎ𝑒 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑝𝑟𝑜𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑖𝑠 𝑑𝑜𝑢𝑏𝑙𝑒𝑑. 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑡ℎ𝑒 𝐺𝑃.
[𝑃𝑅𝑀𝑂 2017]𝐿𝑒𝑡 𝑢,𝑣,𝑤 𝑏𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑖𝑛 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑝𝑟𝑜𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑢 greater than 𝑣 greater than 𝑤. 𝑆𝑢𝑝𝑝𝑜𝑠𝑒 𝑢^40=𝑣^𝑛=𝑤^60 ,𝑓𝑖𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑛.
[𝐽𝐸𝐸 𝑀𝐴𝐼𝑁] 𝐿𝑒𝑡 𝑓: 𝑅 → 𝑅 𝑏𝑒 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑅, (2^(1+𝑥) +2^(1−𝑥) ), 𝑓(𝑥) 𝑎𝑛𝑑 (3^𝑥+3^(−𝑥) ) 𝑎𝑟𝑒 𝑖𝑛 𝐴.𝑃., 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑓(𝑥) 𝑖𝑠 :
[𝐵𝐼𝑇𝑆𝐴𝑇]"Let 1/16 ,a and b be in GP and 1/a,1/b,6 be in AP, where a,b greater than 0. Then, 72(a+b) is equal to"
[𝐽𝐸𝐸 𝑀𝐴𝐼𝑁]𝐹𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑟𝑒 𝑖𝑛 𝐴.𝑃., 𝑤ℎ𝑜𝑠𝑒 𝑠𝑢𝑚 𝑖𝑠 25 𝑎𝑛𝑑 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑖𝑠 2520. 𝐼𝑓 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒 𝑓𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑖𝑠 −1/2, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑟𝑒𝑎𝑡𝑒𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑚𝑜𝑛𝑔𝑠𝑡 𝑡ℎ𝑒𝑚 𝑖𝑠
[JEE MAIN] Let a1, a2, a3,…… a11 be real numbers satisfying a1 = 15, 27 - 2a2 greater than 0 and ak = 2ak-1 - ak-2 for k = 3, 4, …..,11. If [a12 + a22 + …. + a112]/11 = 90 then find the value of [a1 + a2 + …. +a11]/11.
𝐿𝑒𝑡 𝑥,𝑦,𝑧 𝑏𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥^2,𝑦^2,𝑧^2 𝑓𝑜𝑟𝑚 𝑎𝑛 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑝𝑟𝑜𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛. 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 1/(𝑦 +𝑧) ,1/(𝑧 +𝑥) ,1/(𝑥+𝑦) 𝑓𝑜𝑟𝑚 𝑎𝑙𝑠𝑜 𝑎𝑛 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑝𝑟𝑜𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛.