Chapter 11.1, Problem 19WE

a.

To determine

To find: Type of sequence

Answer to Problem 19WE

The given sequence is an Arithmetic Progression (A.P).

Explanation of Solution

Given information: $-3,-1,1,3,\mathrm{.........}?$

From the above sequence the common difference between the two terms are constant. Therefore, the given sequence is an Arithmetic Progression (A.P). Calculation:

Let, first term, $a_1$ = −3

Second term, $a_2$ = −1

Third term, $a_3$ = 1

As we know that, the common difference, $d=a_2-a_1=a_3-a_2$

Now,

  $\begin{array}{l}\Rightarrow d=\left(-1\right)-\left(-3\right)=\left(1\right)-\left(-1\right)\\ \Rightarrow d=-1+3=1+1\\ \Rightarrow d=2=2\end{array}$

Hence, the given sequence is an Arithmetic Progression (A.P)

b.

To determine

To find: Type of sequence

Answer to Problem 19WE

The given sequence is a Geometric progression (G.P).

Explanation of Solution

Given information: $2^{-3},2^{-1},2^1,2^3,\mathrm{...........}?$

From the above sequence the common ratio of the two terms are constant. Therefore, the given sequence is a Geometric progression (G.P).

Calculation:

Let, first term, $a_1$ = $2^{-3}$

Second term, $a_2$ = $2^{-1}$

Third term, $a_3$ = $2^1$

As we know that, the common ratio, $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}$

Now,

  $\begin{array}{l}\Rightarrow r=\frac{2^{-1}}{2^{-3}}=\frac{2^1}{2^{-1}}\\ \Rightarrow r=2^{-1}\times 2^{+3}=2^1\times 2^{+1}\\ \Rightarrow r=2^{-1+3}=2^{1+1}\\ \Rightarrow r=2^2=2^2\end{array}$

Hence, the given sequence is a Geometric progression (G.P).