Question

The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric, and writethe formula for the n" term a, of that sequence.thSequenceTypethn term formula음.Arithmetic(a) 3, 6, 12,...4=ロロローロ0.0GeometricХ5?(b) 3, 8, 13, ...ArithmeticGeometric

Answer 1

Answer:

a) Geometric Sequence

$a_n=3\times2^{n-1}$

b) Arithmetic Sequence

$a_n=a_1+5(n-1)$

Explanation:

a) We have the sequence: 3, 6, 12

For arithmetic sequences, the difference between successive terms are equal:

$\begin{gathered} d=6-3=12-6 \\ d=3=6 \\ \therefore d=3\ne6 \end{gathered}$

This is not an arithmetic sequence

For geometric sequences, the ratio of successive terms are found to be equal:

$\begin{gathered} r=\frac{6}{3}=\frac{12}{6} \\ r=2=2 \\ \therefore r=2=2 \end{gathered}$

This is a geometric sequence

Its formula is thus given as:

$\begin{gathered} a_1=3 \\ r=\frac{12}{6}=2 \\ \text{The general equation of a geometric sequence is given by:} \\ a_n=a_1\cdot r^{n-1} \\ \text{Substituting the values of the known variables into the equation, we have:} \\ a_n=3\times2^{n-1} \\ \\ \therefore a_n=3\times2^{n-1} \end{gathered}$

b) We have the sequence: 3, 8, 13

For arithmetic sequences, the difference between successive terms are equal:

$\begin{gathered} d=8-3=13-8 \\ d=5=5 \\ \therefore d=5=5 \end{gathered}$

This is an arithmetic sequence

Its formula is thus given as:

$\begin{gathered} a_1=3 \\ d=8-3=5 \\ \text{The general formula for arithmetic sequence is given by:} \\ a_n=a_1+(n-1)d \\ \text{Substituting the values of the known variables into the equation, we have:} \\ a_n=3+5(n-1) \\ \\ \therefore a_n=3+5(n-1) \end{gathered}$