Miscellaneous Progressions — XAT Previous-Year Questions

Suppose Haruka has a special key âˆ† in her calculator called delta key:

Rule 1: If the display shows a one-digit number, pressing delta key âˆ† replaces the displayed number with twice its value.

Rule 2: If the display shows a two-digit number, pressing delta key âˆ† replaces the displayed number with the sum of the two digits.

Suppose Haruka enters the value 1 and then presses delta key âˆ† repeatedly.

After pressing the âˆ† key for 68 times, what will be the displayed number?

Consider a_{n + 1} = $\frac{1}{1+{\displaystyle \frac{1}{a_n}}}$ for n = 1, 2, ...., 2008, 2009 where a₁ = 1. Find the value of a₁ a₂ + a₂ a₃ + a₃ a₄ + ... + a₂₀₀₈ a₂₀₀₉ 

An antique store has a collection of eight clocks. At a particular moment, the displayed times on seven of the eight clocks were as follows: 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm and 4:14 pm. If the displayed times of all eight clocks form a mathematical series, then what was the displayed time on the remaining clock?