Processing math: 100%

Properties of Geometric Progression

We will discuss about some of the properties of Geometric Progressions and geometric series which we will frequently use in solving different types of problems on Geometric Progressions.

Property I: When each term of a Geometric Progression is multiplied or divided by a same non-zero quantity, then the new series forms a Geometric Progression having the same common ratio.

Proof:

Let, a1, a2, a3, a4, .................., an, .......... be a Geometric Progression with common r. Then,

an+1an = r, for all n ∈ N ................... (i)

Let k be a non-zero constant. Multiplying all the terms of the given Geometric Progression by k, we obtain the sequence

ka1, ka2, ka3, ka4, ................., kan, ................

Clearly, ka(n+1)kan = a(n+1)an = r for all n ∈ N [Using (i)]

Hence, the new sequence also forms a Geometric Progression with common ratio r.


Property II: In a Geometric Progression the reciprocals of the terms also form a Geometric Progression.

Proof:

Let, a1, a2, a3, a4, .................., an, .......... be a Geometric Progression with common r. Then,

an+1an = r, for all n ∈ N ................... (i)

The series formed by the reciprocals of the terms of the given Geometric Progression is

1a1, 1a2, 1a3, ................., 1an, .................

We have, 1a(n+1)1an = anan+1 = 1r [Using (i)]

So, the new series is a Geometric Progression with common ratio 1r.   

 

Property III: When all the terms of a Geometric Progression be raised to the same power, then the new series also forms a Geometric Progression.

Proof:

Let, a1, a2, a3, a4, .................., an, .......... be a Geometric Progression with common r. Then,

a_(n + 1)/a_n = r, for all n ∈ N ................... (i)

Let k be a non-zero real number. Consider the sequence

a1^k, a2^k, a3^k, ........, an^k, ...........

We have, a_(n +1)^k/a_n^k = (a_(n +1)/a_n)^k = r^k for all n ∈ N, [Using (i)]

Hence, a1^k, a2^k, a3^k, ........, an^k, ........... is a Geometric Progression with common ratio r^k.

Property IV: The product of the first and the last term is always equal to the product of the terms equidistant from the beginning and the end of finite Geometric Progression.

Proof:

Let, a1, a2, a3, a4, .................., an, .......... be a Geometric Progression with common r. Then,

Kth term form the beginning = a_k = a_1r^(k - 1)

Kth term from the end = (n – k + 1)th term form the beginning

                                         = a_(n – k + 1) = a_1r^(n – k)

Therefore, kth term from the beginning)(kth term from the end) = a_ka_(n – k + 1)

= a1r^(k – 1)a1r^(n – k) = a162 r^(n -1) = a1 * a1r^(n – 1) = a1an for all k = 2, 3, ......, n - 1.

Hence, the product of the terms equidistant from the beginning and the end is always same and is equal to the product of the first and the last term.

 

Property V: Three non-zero quantity a, b, c are in Geometric Progression if and only if b^2 = ac.

Proof:

A, b, c are in Geometric Progression ⇔ b/a = c/b = common ratio ⇔ b^2 = ac

Note: When a, b, c are in Geometric Progression, then b is known as the geometric mean of a and c.


Property VI: When the terms of a Geometric Progression are selected at intervals then the new series obtained also a Geometric Progression.

Property VII: In a Geometric Progression of non-zero non-negative terms, then logarithm of each term is form an Arithmetic Progression and vice-versa.

i.e., If a1, a2, a3, a4, .................., an, ..................... are non-zero non-negative terms of a Geometric Progression then loga1, loga2, loga3, loga4, ....................., logan, ......................... forms an Arithmetic Progression and vice-versa.

Proof:

If a1, a2, a3, a4, .................., an, ..................... is a Geometric Progression of non-zero non-negative terms with common ratio r. Then,

a_n = a1r^(n -1), for all n ∈ N

⇒ log a_n = log a1 + (n – 1) log r, for all n ∈ N

Let b_n = log a_n = log a1 + (n – 1) log r, for all n ∈ N

Then, b_ n +1 – b_n = [loga1 + n log r] – [log a1 + (n -1) log r] = log r, for all n ∈ N.

Clearly, b_n + 1 – b_n = log r = constant for all n ∈ N. Hence, b1, b2, b3, b4, ................., bn, ....... i.e., log a1, log a2, log a3, log a4, ..................., log an, ........... be an Arithmetic Progression with common difference log r.

Conversely, let log a1, log a2, log a3, log a4, ..................., log an, ........... be an Arithmetic Progression with common difference d. Then,

log a _(n + 1) – log an = d, for all n ∈ N.

⇒ log (a_n +1/an) = d, for all n ∈ N.

⇒ a_n +1/an = e^d, for all n ∈ N.

⇒ a1, a2, a3, a4, .................., an, ..................... is a Geometric Progression with common ratio e^d.

 Geometric Progression




11 and 12 Grade Math

From Properties of Geometric Progression to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. 5th Grade BODMAS Rule Worksheet | PEMDAS | Order of operations|Answers

    Apr 03, 25 02:56 AM

    5th grade bodmas rule worksheet

    Read More

  2. Before and After Video | Math Worksheets on Number | Before and After

    Apr 03, 25 12:44 AM

    before and after number worksheet
    Free math worksheets on numbers before and after help the kids to check how much they are good at numbers. The purpose of this math activity is to help your child to say a number in order and also hel

    Read More

  3. Order of Numbers Video | Before and After Numbers up to 10 | Counting

    Apr 03, 25 12:39 AM

    We will learn the order of numbers in a number line. In numbers and counting up to 10 we will learn to find the before and after numbers up to 10.

    Read More

  4. Counting Before, After and Between Numbers up to 10 Video | Counting

    Apr 03, 25 12:36 AM

    Before After Between
    Counting before, after and between numbers up to 10 improves the child’s counting skills.

    Read More

  5. Numbers and Counting up to 10 | Why do we Need to Learn Numbers?|Video

    Apr 03, 25 12:33 AM

    Learning Numbers
    We will learn numbers and counting up to 10 to recognize the numerals 1 through 10. Counting numbers are very important to know so that we can understand that numbers have an order and also be able to

    Read More