Three terms of an arithmetic sequence are shown:
\[ f(1) = 3, \quad f(3) = 6, \quad f(7) = 12 \]
What recursive formula could define the sequence?
A. \[ f(n+1) = 2f(n) \]
B. \[ f(n+1) = f(n) + 3 \]
C. \[ f(n+1) = f(n) + 1.5 \]
D. \[ F(n+1) = 0.5f(n) \]
Answer (2)
an a r i t hm e t i c se q u e n ce : d → t h e d i ff ere n ce f ( n ) = f ( 1 ) + ( n − 1 ) ⋅ d f ( n + 1 ) = f ( 1 ) + n ⋅ d f ( n + 1 ) − f ( n ) = f ( 1 ) + n ⋅ d − [ f ( 1 ) + n ⋅ d − d ] = = f ( 1 ) + n d − f ( 1 ) − n d + d = d ⇒ f ( n + 1 ) = f ( n ) + d
f ( 1 ) = 3 an d f ( 3 ) = 6 f ( 3 ) = f ( 1 ) + ( 3 − 1 ) ⋅ d ⇒ 6 = 3 + 2 d ⇒ 2 d = 3 ⇒ d = 1.5 f ( n + 1 ) = f ( n ) + d ⇒ f ( n + 1 ) = f ( n ) + 1.5 A n s . f ( 1 ) = 3 an d f ( n + 1 ) = f ( n ) + 1.5
The recursive formula for the given arithmetic sequence is f ( n + 1 ) = f ( n ) + 1.5 . This formula describes that each term is found by adding 1.5 to the previous term. The correct choice is Option C.
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