Question

The sums 1+2+4, 1+2+4+8, 1+2+4+8+16, … are alternately prime and composite.

Short answer

It is proved that the given assertion is not true in general.

Step-by-step solution

Obtain the numbers are prime and composite

The sum is given by:

$\begin{array}{rcl}{S}_n& =& \underset{k=0}{\overset{n}{\sum 2^k}}\\ & =& \frac{2^{n+1}-1}{2-1}\\ & =& 2^{n+1}-1\\ & & \end{array}$

Where $n=2,3.....$

$2^n-1$is the form of a prime number, and it is called a Mersenne prime. Not all forms of these numbers are prime.

The statement that these numbers are alternately prime and composite is false.

Proof

For example,

Let’s take $n=8$ and $n=9$

$$\begin{gathered} 1+2+2^2+......+2^7=255 \\ 1+2+2^2+......+2^8=511 \end{gathered}$$

Both numbers are composite numbers. Hence it is proved.