Chapter 12.3, Problem 73AYU

To determine

To find: Whether the sequence is arithmetic, geometric or neither, find the common difference or common ratio and also find the sum of the first fifty terms if the sequence is arithmetic or geometric.

Answer to Problem 73AYU

The sequence is neither arithme3tic nor geometric.

Explanation of Solution

Given information:

The series is $1,3,6,\mathrm{10.....}$ .

Calculation:

Here, $a_1=1$ , $a_2=3$ and $a_3=6$ .

Calculate the difference between successive terms.

  $\begin{array}{c}{a}_2-a_1=3-1\\ =2\end{array}$

  $\begin{array}{c}{a}_3-a_2=6-3\\ =3\end{array}$

Since the difference between consecutive terms is not constant the sequence is not arithmetic.

Calculate the common ratio.

  $\begin{array}{c}\frac{a_2}{a_1}=\frac{3}{1}\\ =3\end{array}$

  $\begin{array}{c}\frac{a_3}{a_2}=\frac{6}{3}\\ =2\end{array}$

Since the common ratio of the consecutive terms is not the same the sequence in not geometric.

Therefore, the sequence is neither arithmetic nor geometric.