Chapter 12.3, Problem 98AYU

To determine

To verify: The arithmetic sequence and geometric sequence.

Answer to Problem 98AYU

Yes, a sequence can be both arithmetic and geometric, if it is a constant sequence.

For example, k, k, k,……………….

Explanation of Solution

Given information:

Let, the arithmetic and geometric sequence be: $\begin{array}{l}a,a+d,a+2d,\mathrm{................}\\ a,ar,a{r}^2,\mathrm{........................}\end{array}$

Calculation:

The n^th term of both the sequence should be equal.

Therefore, $\begin{array}{l}a+(n-1)d=a.r^{n-1}\\ \end{array}$

This is only possible when d = 0 and r = 1 as

  $\begin{array}{l}{a}_n=a+(n-1).0=a\\ a_n=a{.1}^{n-1}=a\end{array}$

Therefore, both the sequence become constant sequence.

Hence, a sequence can be both arithmetic and geometric, if it is a constant sequence. (Verified)

For example, k, k, k,……………….