Q. 25 - Calculus

Question

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

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

Step-by-Step Solution

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 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.

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

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

For every even value of k, the term will be one.

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