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?
+128474 The series will be arithmetic
Since the average of the 26th and 28th terms = 94 and the average of the 93rd and 95th terms = 46, we can write :
2a27 = a26 + a28 2a94 = a93 + a95
188 = a26 + a28 92 = a93 + a95
And we have that
a27 = a1 + d ( 26) (1)
a94 = a1 + d (93) (2)
Multiply (1) and (2) through by 2
2a27 = 2a1 + 2d (26)
2a94 = 2a1 + 2d( 93)
188 = 2a1 + 52d (3)
92 = 2a1 + 186d (4)
Subtract ( 4) from (3) and solve for d
96 = -134d
d = 96/-134 = -48/67
Using (1) and subbing to solve for a1
94 = a1 - 48/67 ( 26)
94 + 48/67 (26) = a1
7546 / 67 = a1
