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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 46. What's the first entry?

 Nov 26, 2021
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Using the formula:  tn  =  t1 + (n - 1)d

 

          94  =  t1 + (27 - 1)d                    46  =  t1 + (94 - 1)d

          94  =  t1 + 26d                            46  =  t1 + 93d

 

Combining these two formulas:

                 94  =  t1 + 26d 

                 46  =  t1 + 93d

 

Subtracting down:

                 48  =  - 67d     --->     d  =  -48/67     and     t1  =  112 42/67

 Nov 26, 2021

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