Chapter 12.5, Problem 46E

a.

To determine

To model the given situation in sigma notation.

Answer to Problem 46E

  ${\displaystyle \sum _{n=1}^{\infty }500,000{\left(0.35\right)}^n}$

Explanation of Solution

Given information:

35% convince 35% of 500,000 to buy and this goes on.

We can write the given expression as,

  $500.000{\left(0.35\right)}^n$

(35%=0.35)

Where $n$ takes values from 1 to $\infty$ .

Hence sigma notation is,

  ${\displaystyle \sum _{n=1}^{\infty }500,000{\left(0.35\right)}^n}$

b.

To determine

To determine number of people that will buy the product as a result of advertising campaign.

Answer to Problem 46E

269,230 people will be convinced.

Explanation of Solution

Given information:

35% convince 35% of 500,000 to buy and this goes on.

  ${\displaystyle \sum _{n=1}^{\infty }500,000{\left(0.35\right)}^n}$

Here we have to find sum of this infinite geometric series,

We know $S=\frac{a}{1-r}$

Therefore we get,

  $S=\frac{500,000\left(0.35\right)}{1-0.35}\approx 269,230$

Hence 269,230 people will be convinced.

c.

To determine

To determine the percentage of population convinced.

Answer to Problem 46E

Around 53.8% people convinced

Explanation of Solution

Given information:

269230 people convinced of 500,000 people.

Percentage of population is $\frac{269230}{500000}\times 100\approx 53.8\%$

d.

To determine

To describe the assumption made by advertising agency.

Explanation of Solution

The values we calculated are on assumption that the series goes to infinity.

That is 35% will convince 35% and so on, and does not stops.