Question

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.

$6,12,36,144,...$

Short answer

$6,12,36,144,...$is neither arithmetic nor geometric sequence.

Step-by-step solution

Given information:

Consider the given sequence, $6,12,36,144,...$

Formulas:

$$\begin{gathered} ForArithmeticSequence: \\ Commondifference:d=a_n-a_{n-1} \\ ForGeometricSequence: \\ Commonratio:r=\frac{a_n}{a_{n-1}} \end{gathered}$$

To check whether the given sequence is arithmetic sequence:

To check for arithmetic: Difference of any 2 successive terms is constant, that is common difference $d$.

The given sequence, $6,12,36,144,...$

$$\begin{gathered} d=12-6 \\ =6 \\ d=36-12 \\ =24 \end{gathered}$$

$\therefore$The given sequence is not an arithmetic sequence.

To check whether the given sequence is geometric sequence:

To check for geometric: Ratio of successive terms is constant, that is common ratio $r.$

The given sequence:$6,12,36,144,...$

$$\begin{gathered} r=\frac{12}{6}=2 \\ r=\frac{36}{12}=6 \end{gathered}$$

$\therefore$The given sequence isnot a geometric sequence.