Question

this sequence represents the diameters of circles used to create an art project: 2.5 cm, 3.1 cm, 3.7 cm, 4.3 cm let ( f(n) ) represent diameter in centimeters and ( n ) the term number in the sequence. which equation represents the sequence of diameters? ( f(n)=0.6 n + 1.9 ) ( f(n + 1)=f(n)-0.6 ) ( f(n + 1)=f(n)+1.9 ) ( f(n)=0.6 n + 2.5 )

Explanation:

Step1: Check if it is an arithmetic sequence

Find the common difference \(d\). \(d = 3.1−2.5=0.6\), \(d = 3.7−3.1 = 0.6\), \(d=4.3−3.7 = 0.6\). So it is an arithmetic sequence.

Step2: Analyze recursive formula

For an arithmetic sequence, the recursive formula is \(f(n + 1)=f(n)+d\). Since \(d = 0.6\), \(f(n + 1)=f(n)+0.6\). The formula \(f(n + 1)=f(n)-0.6\) is wrong.

Step3: Analyze explicit formula

The explicit formula for an arithmetic sequence is \(f(n)=f(1)+(n - 1)d\). Here \(f(1)=2.5\), \(d = 0.6\). Then \(f(n)=2.5+(n - 1)\times0.6=0.6n+1.9\). When \(n = 1\), \(f(1)=0.6\times1+1.9=2.5\); when \(n = 2\), \(f(2)=0.6\times2+1.9=3.1\); when \(n = 3\), \(f(3)=0.6\times3+1.9=3.7\); when \(n = 4\), \(f(4)=0.6\times4+1.9=4.3\). The formula \(f(n)=0.6n + 2.5\) is wrong (\(n = 1\), \(0.6\times1+2.5=3.1
eq2.5\)). The formula \(f(n + 1)=f(n)+1.9\) is wrong (\(d
eq1.9\)).

Answer:

\(f(n)=0.6n + 1.9\)