Question

In Problems 67– 80, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.

$1,3,6,10,.....$

Short answer

The given sequence $1,3,6,10,.....$is neither.

Step-by-step solution

Step 1. Write the given information.

The given sequence is:

$1,3,6,10,.....$

Step 2. Find the common difference.

$$\begin{gathered} a_1=1,a_2=3,a_3=6,a_4=10 \\ {a}_2-a_1=3-1;d=2 \\ {a}_3-a_2=6-3;d=3 \end{gathered}$$

As we can see that the common difference is not the same, therefore the given sequence is not an arithmetic sequence.

Step 3. Find the common ratio.

$$\begin{gathered} \frac{a_2}{a_1}=\frac{3}{1};r=3 \\ \frac{a_3}{a_2}=\frac{6}{3};r=2 \end{gathered}$$

As we can see that the common ratio is not the same, therefore the given sequence is not a geometric sequence.

The given sequence is neither arithmetic nor geometric.