How to find common ratio with first and last terms?

It is a branch of mathematics that deals usually with the non-negative real numbers which including sometimes the transfinite cardinals and with the appliance or merging of the operations of addition, subtraction, multiplication, and division. The basic operations that come under arithmetic are addition, subtraction, division, and multiplication. The BODMAS rule is followed to calculate or order any operation involving +, −, ×, and ÷. The order of operation is,

B: Brackets

O: Order

D: Division

M: Multiplication

A: Addition

S: Subtraction

Progression 

Progression may be a list of numbers that shows or exhibit a specific pattern. The most basic difference between a sequence and a progression is that to calculate its n^th term, a progression has a specific or fixed formula i.e. T_n = a + (n-1)d  which is the formula of the n^th term of an arithmetic progression.

Geometric Progression 

A geometric progression is a sequence where every term holds a constant ratio to its previous term. The common ratio represented as “r” remains the same for all consecutive terms in a particular GP. 

Finding Common ratio with first and last terms

The common ratio is the amount between each number in a geometric sequence. It is called the common ratio because it is the same to each number or common, and it also is the ratio between two consecutive numbers i.e, a number divided by its previous number in the sequence.

The last term is simply the term at which a particular series or sequence line arithmetic progression or geometric progression ends or terminates. It is generally denoted by small ‘l’, First term is the initial term of a series or any sequence like arithmetic progression, geometric progression harmonic progression, etc. It is generally denoted with small ‘a’ and Total terms are the total number of terms in a particular series which is denoted by ‘n’.

It is  known that,

l = a × r ^(n-1)

l/a = r ^(n-1)

(l/a)^(1/(n-1)) = r 

With this formula, calculate the common ratio if the first and last terms are given. Let’s look at some examples to understand this formula in more detail,

Sample Problems

Question 1: In a G.P first term is ‘1’ and 4th term is ‘ 27’ then find the common ratio of the same.

Solution:

Here a = 1 and a4 = 27 and let common ratio is  ‘r’ . So

⇒ a4 = a × ( r^4-1)

⇒ 27 =  1 × r^4-1 = r³

⇒ Common ratio =  r = 3

Question 2: The 1st term of a geometric progression is 64 and the 5th term is 4. If the sum of all terms is 128, what is the common ratio?

Solution:

Since the 1st term is 64 and the 5th term is 4,

It is obvious that successive terms decrease in value. 

Therefore, r < 1. 

So, the sum of all terms is a/(1 – r) = 128.

 Solving, we get r = 1/2.

Question 3: The product of the first three terms of a geometric progression is 512. If 2 is added to its second term, the three terms form an A. P. Find the terms of the geometric progression.

Solution:

Let the first three terms of G.P. are ,a,ar

Given that a × a × a = 512 ⇒  a³ = 512 ⇒ a = 8

Now, a+2 are in A.P.

⇒ 8r² + 8 = 20r

⇒ 8r² – 20r + 8 =0

⇒ 2r² – 5r + 2 =0

⇒ 2r²– 4r – r + 2 =0

⇒ 2r(r-2) – (r-2) =0

⇒ (2r-1)(r-2) = 0

⇒ r = 2 or 1/2

When r = 2, the terms are 4, 8, 16

When r = 1/2, then the terms are 16, 8, 4.

Question 4: Is the following series a geometric progression? 

5 , 20 , 80 , 320 , …

Answer:

Yes , it is an geometric progression with common ratio 4.

Question 5: Can a common ratio be a fraction of a negative number?

Answer:

Yes , common ratio can be a fraction or a negative number .