Question

the first term in an arithmetic sequence is 5. the fourth term in the sequence is -4. the tenth term is -22. which function can be used to find the nth term of the arithmetic sequence? a) f(n) = -n b) f(n) = n + 4 c) f(n) = -3n + 8 d) f(n) = \frac{1}{2}(n + 5) + 2

Explanation:

Step1: Recall Arithmetic Sequence Formula

The formula for the \(n\)-th term of an arithmetic sequence is \(f(n)=a_1+(n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Given \(a_1 = 5\).

Step2: Find Common Difference \(d\)

We know the fourth term \(a_4=-4\). Using the formula \(a_n=a_1+(n - 1)d\), for \(n = 4\):
\(-4=5+(4 - 1)d\)
\(-4-5 = 3d\)
\(-9=3d\)
\(d=-3\)

Step3: Derive the \(n\)-th Term Function

Substitute \(a_1 = 5\) and \(d=-3\) into the formula:
\(f(n)=5+(n - 1)(-3)\)
\(f(n)=5-3n + 3\)
\(f(n)=-3n+8\)
We can also verify by plugging \(n = 1\), \(n = 4\), \(n = 10\) into each option:

• For option a: \(f(1)=-1

eq5\), so a is wrong.

• For option b: \(f(1)=1 + 4=5\), \(f(4)=4 + 4=8

eq-4\), so b is wrong.

• For option c: \(f(1)=-3(1)+8 = 5\), \(f(4)=-3(4)+8=-4\), \(f(10)=-3(10)+8=-22\), which matches the given terms.
• For option d: \(f(1)=\frac{1}{2}(1 + 5)+2=3 + 2=5\), \(f(4)=\frac{1}{2}(4 + 5)+2=\frac{9}{2}+2=\frac{13}{2}

eq-4\), so d is wrong.

Answer:

c. \(f(n)=-3n + 8\)