IOQM Engineering Entrance Progressions Part2

[𝐵𝐼𝑇𝑆𝐴𝑇] 𝐼𝑛 𝑎 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑜𝑓 21 𝑡𝑒𝑟𝑚𝑠, 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 11 𝑡𝑒𝑟𝑚𝑠 𝑎𝑟𝑒 𝑖𝑛 𝐴𝑃 𝑤𝑖𝑡ℎ 𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 2 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑙𝑎𝑠𝑡 11 𝑡𝑒𝑟𝑚𝑠 𝑎𝑟𝑒 𝑖𝑛 𝐺𝑃 𝑤𝑖𝑡ℎ 𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 2. 𝐼𝑓 𝑡ℎ𝑒 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑜𝑓 𝐴𝑃 𝑏𝑒 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝐺𝑃, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑛𝑡𝑖𝑟𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑖𝑠

[𝐵𝐼𝑇𝑆𝐴𝑇] 𝐼𝑓 𝑎 + 2𝑏 + 3𝑐 = 12, (𝑎, 𝑏, 𝑐 ∈𝑅+), 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑏2𝑐3 𝑖𝑠

"[BITSAT] Sum of n terms of the infinite series 1x32 + 2x52 + 3x72 + ..... ∞ is"

𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑢𝑚 ∑_(𝑘=1)^∞▒∑_(𝑛=1)^∞▒𝑘/2^(𝑛+𝑘) 𝑖𝑠

"After striking the floor certain ball rebounds 4/5th of its height from which it has fallen. Calclulate the total distance travelled by a ball before coming to rest if it is dropped from 120m height. "

"If loga,logb, and logc are in A.P. and also loga−log2b, log2b−log3c, log3c−loga are in A.P., then"
(a) a,b,c are in H.P (b) a, 2b, 3c are in A.P. ( c) a, b, c are side of triangle (d) None of the above

(666…6)^2+(888…8)= ?[Each number is n digits]

𝑖𝑓 3+1/4 (3+𝑑)+1/4^2 (3+2𝑑)+….∞=8, 𝑡ℎ𝑒𝑛 𝑑= ?

[𝑁𝑇𝑆𝐸] 𝑇ℎ𝑒 𝑜𝑑𝑑 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 ℎ𝑎𝑣𝑒 𝑏𝑒𝑒𝑛 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑖𝑛𝑡𝑜 𝑔𝑟𝑜𝑢𝑝𝑠 𝑎𝑠 (1,3);(5,7,9,11); (13,15,17,19,21,23);….𝑇ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑖𝑛 10𝑡ℎ 𝑔𝑟𝑜𝑢𝑝 𝑖𝑠 ?

[𝐴𝑀𝐶 2005]𝐼𝑛 𝑡ℎ𝑒 𝑓𝑖𝑣𝑒−𝑠𝑖𝑑𝑒𝑑 𝑠𝑡𝑎𝑟 𝑠ℎ𝑜𝑤𝑛, 𝑡ℎ𝑒 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝐴, 𝐵, 𝐶, 𝐷, 𝑎𝑛𝑑 𝐸 𝑎𝑟𝑒 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑑 𝑏𝑦
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 3, 5, 6, 7, 𝑎𝑛𝑑 9, 𝑎𝑙𝑡ℎ𝑜𝑢𝑔ℎ 𝑛𝑜𝑡 𝑛𝑒𝑐𝑒𝑠𝑠𝑎𝑟𝑖𝑙𝑦 𝑖𝑛 𝑡ℎ𝑖𝑠 𝑜𝑟𝑑𝑒𝑟. 𝑇ℎ𝑒 𝑠𝑢𝑚𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑒𝑛𝑑𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑠𝑒𝑔𝑚𝑒𝑛𝑡𝑠 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, 𝐷𝐸, 𝑎𝑛𝑑 𝐸𝐴 𝑓𝑜𝑟𝑚 𝑎𝑛 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒, 𝑎𝑙𝑡ℎ𝑜𝑢𝑔ℎ 𝑛𝑜𝑡 𝑛𝑒𝑐𝑒𝑠𝑠𝑎𝑟𝑖𝑙𝑦 𝑖𝑛 𝑡ℎ𝑖𝑠 𝑜𝑟𝑑𝑒𝑟. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒?

𝑇ℎ𝑒 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑎𝑛 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑎𝑑𝑑 𝑡𝑜 715. 𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑖𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑏𝑦 1, 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑏𝑦 3, 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑏𝑦 5, 𝑎𝑛𝑑 𝑖𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙, 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑡𝑒𝑟𝑚 𝑖𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑘𝑡ℎ 𝑜𝑑𝑑 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟. 𝑇ℎ𝑒 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑒𝑤 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑎𝑑𝑑 𝑡𝑜 836.
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡, 𝑙𝑎𝑠𝑡, 𝑎𝑛𝑑 𝑚𝑖𝑑𝑑𝑙𝑒 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒. [AIME 2012]