Problems on Geometric Progression

Here we will learn how to solve different types of problems on Geometric Progression.

1. Find the common ratio of the Geometric Progression whose, sum of the third and fifth terms is 90 and its first term is 1.

Solution:

The first term of the given Geometric Progression a = 1.

Let ‘r’ be the common ratio of the Geometric Progression.

According to the problem,

 t_3 + t_5 = 90

ar^2 + ar^4 = 90

r^2 + r^4 = 90

r^4 + r^2 – 90 = 0

r^2 + 10r^2 - 9r^2 - 90 = 0

(r^2 + 10)(r^2 - 9) =0

r^2 - 9 = 0

r^2 = 9

r = ±3

Therefore, the common ratio of the Geometric Progression is -3 or 3.

2. Find a Geometric Progress for which the sum of first two terms is -4 and the fifth term is 4 times the third term.

Solution:

Let ‘a’ be the first term and ‘r’ be the common ratio of the given Geometric Progression.

Then, according to the problem the sum of first two terms is -4

t_1 + t_2 = -4

a + ar = -4 .................. (i)

and the fifth term is 4 times the third term.

t_5 = 4t_3

ar^4 = 4ar^2

r^2 = 4

r = ±2

Putting r = 2 and -2 respectively in (i), we get a = -4/3 and a = 4.

Thus, the required Geometric Progression is -4/3, -8/3, -16/3, ............ or 4, -8, 16, -32, ........................


3. Prove that in a Geometric Progression of finite number of terms the product of any two terms equidistant from the beginning and the end is constant and is equal to the product of the first and last and last terms.

Solution:

Let ‘a’ be the first term, ‘b’ the last term and ‘r’ the common ratio of a finite Geometric Progression.

Then the nth term from the beginning = a* r^(n - 1)

And the nth term from the end = b/r^(n -1)

Therefore, the product of two equidistant terms from the beginning and end (i.e., the terms at the nth positions) = a * r^(n - 1) * b/r^(n -1)  = a * b = constant = first term X last term.          Proved.

 Geometric Progression



11 and 12 Grade Math 

From Problems on Geometric Progression to HOME PAGE


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.



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.



Share this page: What’s this?

Recent Articles

  1. Multiplication of a Number by a 3-Digit Number |3-Digit Multiplication

    Mar 28, 24 06:33 PM

    Multiplying by 3-Digit Number
    In multiplication of a number by a 3-digit number are explained here step by step. Consider the following examples on multiplication of a number by a 3-digit number: 1. Find the product of 36 × 137

    Read More

  2. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  3. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  4. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  5. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More