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There is a sequence of numbers such that every entry except for the first entry is the arithmetic mean of its two neighboring entries. The 27th entry is 94 and the 94th entry is 27. What's the first entry?


Thanks in advance!

 Jan 7, 2019

94 - 27 = 67, so between the 27th and 94th entries, there are 67-1 = 66 entries. The entries before the 94th entry were probably one greater than the entry after, so entry 93 is 28, entry 92 is 29, etc. This makes sense because the average of E92 and E94 is 28, which is the value in E93. 


If the 27th entry is 94, then the 26th entry must be 95, the 25th entry must be 96, etc. From the 27th entry, there are 26 entries ahead of it, so the finrst entry is 94 + 26 = 120. The first entry is \(\boxed{120}\).


Hope this helps, 

- PM

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