Sequences and Series

Hong Kong

Stage 4 - Stage 5

Lesson

When a quantity keeps changing in size across fixed length time intervals, so that at the beginning of each new interval the new size is some constant factor multiplied by the previous size, then we say that the change in quantity is geometric, and the sequence of sizes forms a geometric progression with the common ratio equal to that constant factor.

Maybe this will be more clear in an example. Suppose I spend $10%$10% of my $\$500$$500 birthday money every week. At the beginning of the second week, I'll have spent $\$50$$50, and will only have $\$450$$450 or $90%$90% of the birthday money left. At the beginning of the third week I'll spend slightly less - $\$45$$45 or $10%$10% so at the beginning of the third week $\$405$$405 will remain. The sequence of birthday dollars that remain at the beginning of the first five weeks is given as $500,450,405,364.5,328.05$500,450,405,364.5,328.05 and we can see here that this sequence is geometric with the first term $a=500$`a`=500 and the common ratio $r=0.9$`r`=0.9.

Now suppose that instead of spending $10%$10% of my birthday money, I decide on a plan to double my money using the following strategy. Firstly, I'll add $10%$10% to the birthday money, so that at the beginning of the second week, the additional $\$50$$50 will take the total to $\$550$$550. At the beginning of week $3$3 I'll add a further $10%$10% of this accrued $\$550$$550, so that the new total becomes $\$605$$605. I'll keep adding $10%$10% of what ever is there at the beginning of each week until I exceed $\$1000$$1000.

The geometric progression for this plan becomes $500,550,605,665.5,732.05,...$500,550,605,665.5,732.05,...

This progression has $a=500$`a`=500 and $r=\frac{550}{500}=1.1$`r`=550500=1.1, so that the $n$`n`th term of the progression is given by $t_n=500\left(1.1\right)^{n-1}$`t``n`=500(1.1)`n`−1.

By setting $500\left(1.1\right)^{n-1}=1000$500(1.1)`n`−1=1000 and dividing by $500$500, I know that I have to find $n$`n` so that:

$\left(1.1\right)^{n-1}>2$(1.1)`n`−1>2

I could guess and check for $n$`n` and would eventually come up with $n=8$`n`=8. But an algebraic way to proceed is to take logarithms so that:

$\left(n-1\right)\log_{10}\left(1.1\right)>\log_{10}\left(2\right)$(`n`−1)`l``o``g`10(1.1)>`l``o``g`10(2)

By dividing by $\log_{10}\left(1.1\right)$`l``o``g`10(1.1) and adding one to both sides, we have:

$n-1>\frac{\log_{10}\left(2\right)}{\log_{10}\left(1.1\right)}$`n`−1>`l``o``g`10(2)`l``o``g`10(1.1)

$\therefore n>\frac{\log_{10}\left(2\right)}{\log_{10}\left(1.1\right)}+1$∴`n`>`l``o``g`10(2)`l``o``g`10(1.1)+1

Using a scientific calculator I see that $n$`n` must be greater than $8.27..$8.27.., so after $9$9 weeks my birthday money will exceed $\$1000$$1000.

As a final example, suppose I start with my $\$500$$500, and add to it $90%$90% of $\$500$$500 in the second week. That's an addition of $\$450$$450 and the total becomes $\$950$$950. Then I add $90%$90% of the $\$450$$450 or $\$405$$405 to the new total to bring the savings to $\$1355$$1355. Again I add $90%$90% of $\$405$$405 or $\$364.50$$364.50 to bring the total savings to $\$1719.50$$1719.50. If I keep the pattern of savings going, how much will I have?

The total savings $S$`S` in dollars is given by the geometric sum:

$S=500+0.9\left(500\right)+0.9\left(0.9\left(500\right)\right)+0.9\left(0.9\left(0.9\left(500\right)\right)\right)+...$`S`=500+0.9(500)+0.9(0.9(500))+0.9(0.9(0.9(500)))+...

In other words,

$S=500+405+364.50+...$`S`=500+405+364.50+...

If I continue saving in this manner forever, then, because $S$`S` is a geometric sum with a ratio between $0$0 and $1$1, then we know it has a limiting sum. No matter how many weeks I continue the savings pattern, I will never exceed $\$5000$$5000, as shown by the following calculation.

$S_{\infty}=\frac{a}{1-r}=\frac{500}{1-0.9}=5000$`S`∞=`a`1−`r`=5001−0.9=5000

Average annual salaries are expected to increase by $5$5 percent each year. If the average annual salary this year is found to be $\$49000$$49000:

Calculate the expected average annual salary in $4$4 years, correct to the nearest cent.

This year, Aaron starts at a new job in which he will receive the average annual salary for each year of his employment. Over the coming $4$4 years (including this year) he plans to save half of each year’s annual salary.

What will be his total savings over these $4$4 years? Give your answer correct to the nearest cent.

To test the effectiveness of a new antibiotic, a certain bacteria is introduced to a body and the number of bacteria is monitored. Initially, there are $19$19 bacteria in the body, and after four hours the number is found to double.

If the bacterial population continues to double every four hours, how many bacteria will there be in the body after $24$24 hours?

The antibiotic is applied after $24$24 hours, and is found to kill one third of the germs every two hours. How many bacteria will there be left in the body $24$24 hours after applying the antibiotic? Assume the bacteria stops multiplying and round your answer to the nearest integer if necessary.

The first blow of a hammer drives a post a distance of $144$144 cm into the ground. Each successive blow drives the post $\frac{5}{6}$56 as far as the preceding blow. In order for the post to become stable, it needs to be driven $\frac{4651}{9}$46519 cm into the ground.

If $n$`n` is the number of hammer strikes needed for the pole to become stable, find $n$`n`.