Question

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{0,1,0,1,0,1,\dots \right\}$

Short answer

The plausible formula for the general term of the sequence is$a_k=\frac{1}{2}\left[1+{\left(-1\right)}^k\right].$

Step-by-step solution

Step 1. Given information.

The given sequence is$\left\{0,1,0,1,0,1,\dots \right\}.$

Step 2. plausible formula for the general term. 

the sequence $\left\{0,1,0,1,0,1,\dots \right\}$has $0\&1$alternately.

the sequence can be written as follows.

role="math" localid="1649186161903" ${\left\{\frac{1}{2}\left[1+{\left(-1\right)}^k\right]\right\}}_{k=1}^{\infty }=\left\{0,1,0,1,0,1,\dots \right\}$

where for every odd value of k,the term will be zero

For every even value ofk,the term will be one.

So plausible formula for the general term of the sequence is role="math" localid="1649186184028" $a_k=\frac{1}{2}\left[1+{\left(-1\right)}^k\right]$