Geometric Sequence and Series

Explanation: First let us work with the sequence;

To identify each team in the sequence we’re going t o use U₁, U₂, U₃ etc. We know that U₁ = 4, U₂ = 20, U₃ = 100. To work out the common ratio we divide the following term by the previous term ie.

For the sequence above the common ratio is;

The next sequence was;
The same principal applies. To find the common ratio we use;

Plug in the numbers as shown below;

Explanation
Find the 10th and nth terms in the sequence;

Let us start with the first sequence.

For the above sequence we know that a = 3 and the common ratio r is;

To find the 10th term we use the formula described above so that;

We know n = 10, a=3, and r=2 so;


Next we shall look at the next sequence shown below;

Again here to find the 10th term we use the formula …we know that for the sequence a=40, r= -½ and n=10, so we get

Again to find the nth term formula we use what we know to get;

Here are more examples involving geometric sequences; Suppose a positive geometric sequence is 4 and the 4th term is 8. What is the common ratio and the first and 10th terms.
First let us find the common ratio, we use what we know; we know that;

…and the…

…if we divide both expressions to get;

So the common ratio is √2.
Next we shall find the first term. We can use the common ratio found above so we substitute in one of the equations above since a represents the first term.

Next we shall find the 10th term. We can use the geometric sequence formula; we know a=2√2, r=√2 and n=10, therefore;

…therefore the 10th term is 64.

ExplanationFirst let us find the possible values of x. We know that;

This is the common ratio if it is a geometric sequence so we have;

…simplify to get…

…simplify further to get;

…we need to solve to find x, factorise to get;

…since the sequence contain only positive numbers we shall accept x=6.
Next we shall find the 10th term, …simply we use the formulas;

…we know that r=2, a=3 and n=10, so we get;

…therefore the 10th term is 1536.

ExplanationIf the total increases by 4% each year from the previous year that must mean that every year the total will 100% from the previous year plus 4%, therefore.

Converting that into decimals gives.

That means we will use the multiplication factor 1.04 after each year for the rise. After the first year the investment is worth;

…and after 2 years it will be worth.

It is therefore logical to conclude that after 5 years the investment will be worth;

The above expression is a geometric sequence formula where a =£A and r=1.04.
Above it is written that investment is worth £20,000, we can use this with the formula to find the initial investment; We form an expression like this;

…therefore;
So the initial investment was £16438.54.
We can use this to work out what the investment will be worth after 10 years; …we simply substitute A in the equation;

So the investment will be worth £ 24333.06.

Explanation
This example involves inequalities we can use the nth term formula;

We want the nth term to be greater than 1 million that is;

…therefore…

…we simplify;

…we must use logs here to solve n;
< …therefore n is 20 which represents the 10th term the first to exceed 1000000.

Explanation
For the first series;

…we have to find up to 20 terms, we know that a=2, r-3 and n=20, because r=3 is greater than 1. We shall use the formula;

For the next series we will have to find the position of the last term;

…we know that a=1024 and r= -½ when nth term = 1

…we solve to find n

…because (-2)^n-1 = (-1)^n-1(2^n-1) = 1024, that must mean that; (-1)^n-1 must be positive and…

…now we can use logs to solve n;

Now we can find the sum of the series according to n=11. Because r=-½ is less than 1 we use the formula;

…so we have;

Explanation: Be careful here with the question. The investor invests £4000 every new year and the total amount for each year is calculated on the 31st of Decemeber.
By now we know that a rate of 4% means x1.04. At the end of year 1, the amount is;

at the start of year 2 the amount is;

…notice that we have added the new investment at the end and at the end of year 2 the amount is;

…notice that at the end of each year the total amount is multiplied by 1.04.
At the start of year 3 the amount;

…that must mean that at the end of year 10, the amount is;

The part in brackets above is a geometric series and therefore we can use the formula for it;

…we know that a=1.04, r=1.04 and n=10, therefore;

Explanation: …we know that a=1, r=2, we can substitute this into the formula;

…therefore the sum to nth terms is;

…the inequality we’re after is;



…we can now use logs to find the value of n;

…we can conclude that the sum of the series needs 21 terms for it to exceed 2000000.