Question

In the problem, show that each sequence is arithmetic. Find the common difference and write out the first four terms.

$\left\{s_n\right\}=\left\{e^{\ln n}\right\}$

Short answer

The solution is $d=1$; first four terms$=1,2,3,4$.

Step-by-step solution

Step 1. Given information 

$\left\{s_n\right\}=\left\{e^{\ln n}\right\}$

Step 2. Calculation 

The first term is$s_1=e^{\ln 1}=1$

The $n$th term and the $\left(n-1\right)$st term of the sequence are-

$s_n=e^{\ln n}$ and role="math" localid="1649312659393" $s_{n-1}=e^{\ln \left(n-1\right)}$

The common difference is -

$d=s_n-s_{n-1}=\left(e^{\ln n}\right)-\left[e^{\ln \left(n-1\right)}\right]=n-\left(n-1\right)=1$

Since the difference of any two successive terms is the constant $1$, the sequence $\left\{s_n\right\}$is arithmetic.

The first four terms are$1,2,3,4$.