Geometric progression. Example with solution

Consider a series.

7 28 112 448 1792 ...

It is quite clear that the value of any of its elements is four times greater than the previous one. Hence, this series is a progression.

Geometric progression Called an infinite sequence of numbers, the main feature of which is that the next number is obtained from the previous one by multiplying by a certain number. This is expressed by the following formula.

A _{z +1} = a _z · q, where z is the number of the selected element.

Correspondingly, z ∈ N.

The period when the geometric progression is studied in school is grade 9. Examples will help you understand the concept:

0.25 0.125 0.0625 ...

18 6 2 ...

Proceeding from this formula, the denominator of the progression can be found as follows:

Neither q nor b _z can be equal to zero. Also, each of the elements of the numerical sequence of the progression should not be zero.

Accordingly, to find the next number of the series, we must multiply the last by q.

To specify this progression, you must specify its first element and denominator. After that, it is possible to find any of the subsequent members and their sum.

Varieties

Depending on q and a _1, this progression is divided into several types:

• If both a ₁ and q are greater than one, then such a sequence is a geometric progression increasing with each next element. An example of this is presented below.

Example: a ₁ = 3, q = 2 - both parameters are greater than one.

Then the numerical sequence can be written as:

3 6 12 24 48 ...

• If | q | Less than one, that is, multiplication by it is equivalent to division, then a progression with similar conditions is a decreasing geometric progression. An example of this is presented below.

Example: a ₁ = 6, q = 1/3 - a ₁ more than one, q - less.

Then the numerical sequence can be written thus:

6 2 2/3 ... - any element is greater than the element following it, 3 times.

• It is alternating. If q <0, then the signs of the sequence numbers constantly alternate regardless of a ₁ , and the elements neither increase nor decrease.

Example: a ₁ = -3, q = -2 - both parameters are less than zero.

Then the numerical sequence can be written as:

-3, 6, -12, 24, ...

Formulas

For the convenient use of geometric progressions, there are many formulas:

• The formula of the zth term. Allows you to calculate an element that is under a specific number without calculating the previous numbers.

Example: q = 3, a ₁ = 4. It is required to calculate the fourth element of the progression.

The solution is: a ₄ = 4 · 3 ^4-1 = 4 · 3 ³ = 4 · 27 = 108.

• The sum of the first elements, whose number is Z. Allows to calculate the sum of all elements of the sequence up to a _z inclusive.

Since (1 - q ) stands in the denominator, then (1 - q) ≠ 0, therefore, q is not equal to 1.

Note: if q = 1, then the progression would be a series of infinitely repeating numbers.

The sum of the geometric progression, examples: a ₁ = 2, q = -2. Calculate S ₅ .

Solution: S ₅ = 22 - calculation by formula.

• The sum, if | Q | <1 and if z tends to infinity.

Example: a ₁ = 2 _, q = 0.5. Find the sum.

Solution: S _z = 2 · = 4

If you count the sum of several members manually, you can see that it really strives for four.

S _z = 2 + 1 + 0.5 + 0.25 + 0.125 + 0.0625 = 3.9375 4

Some properties:

• Characteristic property. If the following condition Is satisfied for any z , then the given numerical series is a geometric progression:

A _z ² = A _{z -1} · A _{z + 1}

• Similarly, the square of any number of a geometric progression is found by adding the squares of two other any numbers in a given series if they are equidistant from this element.

And _z ² = a _{z - t} ² + a _{z + t} ² , where t is the distance between these numbers.

• Elements differ by a factor of q .
• Logarithms of the elements of the progression also form a progression, but already arithmetic, that is, each of them is greater than the previous one by a certain number.

Examples of some classical problems

To better understand what a geometric progression is, examples with a solution for class 9 can help.

• Conditions: a ₁ = 3, a ₃ = 48. Find the q .

Solution: each successive element is greater than the previous one in q time. It is necessary to express some elements through others using the denominator.

Therefore, a ₃ = q ² · a ₁

With the substitution q = 4

• Conditions: a ₂ = 6, a ₃ = 12. Calculate S ₆ .

Solution: For this it is sufficient to find q, the first element and substitute in the formula.

A ₃ = q · a ₂ , therefore, q = 2

A ₂ = q · A ₁ , therefore A ₁ = 3

S ₆ = 189

• · A ₁ = 10, q = -2. Find the fourth element of the progression.

Solution: for this it is sufficient to express the fourth element through the first and through the denominator.

A ₄ = q ³ · a ₁ = -80

Example of application:

• The client of the bank made a contribution to the amount of 10,000 rubles, under the terms of which each year to the client to the principal amount will be added 6% of it. How much money will be in the account in 4 years?

Decision: The initial amount is 10 thousand rubles. Hence, a year after the investment in the account there will be an amount equal to 10000 + 10000 · 0.06 = 10000 · 1.06

Accordingly, the amount on the account in another year will be expressed as follows:

(10000 · 1.06) · 0.06 + 10,000 · 1.06 = 1.06 · 1.06 · 10,000

That is, every year the amount is increased by 1.06 times. So, in order to find the amount of funds on the account in 4 years, it is enough to find the fourth element of the progression, which is set by the first element equal to 10 thousand, and the denominator equal to 1.06.

S = 1.06 · 1.06 · 1.06 · 1.06 · 10000 = 12625

Examples of tasks for calculating the sum:

In various problems, a geometric progression is used. An example of finding a sum can be specified as follows:

A ₁ = 4, q = 2, calculate S ₅ .

Solution: all the data necessary for the calculation are known, you just need to substitute them in the formula.

S ₅ = 124

• A ₂ = 6, a ₃ = 18. Calculate the sum of the first six elements.

Decision:

In geom. Progression, each next element is greater than the previous one by q times, that is, to calculate the sum it is necessary to know the element a ₁ and the denominator q .

A ₂ · Q = a ₃

Q = 3

Similarly, it is required to find a ₁ , knowing a ₂ and q .

A ₁ · Q = a ₂

A ₁ = 2

And further it is enough to substitute the known data in the formula of the sum.

S ₆ = 728.