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# find the fifth term of the geometric progression whose first term is 2 and common ratio is 3.

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## The fifth term of a G.P. is 1875 . If the first term is 3 , find the common ratio

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Question

## The fifth term of a G.P. is 1875. If the first term is 3, find the common ratio

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Updated on : 2022-09-05

It is given that the first term of the G.P is a=3 and also the 5th term is T

Solution Verified by Toppr 5 ​ =1875.

We know that the general term of an geometric progression with first term a and common ratio r is T

n ​ =ar n−1 , therefore, T 5 ​ =ar 5−1 =ar 4

Now substitute a=3,n=5 and T

5 ​ =1875 in T 5 ​ =ar 4 as follows: T 5 ​ =ar 4 ⇒1875=3r 4 ⇒r 4 = 3 1875 ​ ⇒r 4 =625 ⇒r 4 =5 4 ⇒r=5

Hence the common ratio is 5.

20 0

स्रोत : www.toppr.com

## Nth term of GP

How to find the nth term of geometric progression? Learn more about the nth term of a gp with solved examples and interactive questions the Cuemath way!

## nth term of a GP

Hey kids! Meet Sam's family.

Here is his family tree.

Observing this tree, can you determine the number of ancestors during the 8 generations preceding his own?

You will have to use the concept of geometric sequence to answer this.

Don't worry! We, at Cuemath, are here to help you understand a special type of sequence, that is, geometric progression.

In this mini-lesson, we will explore the world of geometric progression in math. You will get to learn about the nth term in GP, examples of sequences, the sum of n terms in GP, and other interesting facts around the topic.

## Lesson Plan

1. What Is Meant By Geometric Progression?

2. Important Notes

3. Solved Examples on Geometric Progression

4. Thinking Out of the Box!

5. Interactive Questions on Geometric Progression

## What Is Meant by Geometric Progression?

A geometric sequence is a sequence where every term bears a constant ratio to its preceding term.

### Example

Consider an example of geometric sequence: 1, 4, 16, 64, ...

Observe that 4 1 = 16 4 = 64 16 =4 4 1 = 16 4 = 64 16 = 4

Here, 4 is a common ratio.

## How To Determine the nth Term of GP?

Let the first term of the sequence be a

a

and the common ratio be r

r . Then, nth n th

term of the sequence is given by:

an=arn−1

Use the below simulation to calculate the nth term of some geometric progressions.

Input the values of n

n

for the number of terms you want to calculate.

### Geometric Progression: Sum

Consider a geometric progression whose first term is a

a

and the common ratio is r

r .

The sum of the first n

n

terms of a geometric progression is:

Sn= a(1−rn) 1−r S n = a ( 1 − r n ) 1 − r

### Example

Can you calculate the nth term of the geometric progression if the first two terms are 10 and 20?

First, you need to calculate the common ratio r

r

of the geometric series by dividing the second term by the first term.

r = 20 10 =2

Then substitute the values of the first term a

a

and the common ratio r

r

into the formula of the nth term of the geometric progression an=arn−1

a n = a r n − 1 . an =arn−1 =10(2)n−1 =10 2n 2 =5(2)n Important Notes

In a geometric progression, each successive term is obtained by multiplying the common ratio to its preceding term.

The formula for the nth

n t h

term of a geometric progression whose first term is a

a

and common ratio is r

r is: an=arn−1 a n = a r n − 1

The sum of n terms in GP whose first term is a

a

and the common ratio is r

r

can be calculated using the formula:

Sn= a(1−rn) 1−r

## Solved Examples

Example 1

Look at the pattern shown below.

Observe that each square is half of the size of the square next to it.

Which sequence does this pattern represent?

### Solution

Let's write the sequence represented in the figure.

1, 1 2 , 1 4 , 1 8 , 1 16 ,...

Every successive term is obtained by dividing its preceding term by 2

The sequence exhibits a common ratio of

1 2 1 2 ∴ ∴

The pattern represents the geometric progression.

Example 2

Hailey's teacher asks her to find the 10th

10 th

term of the sequence: 1, 3, 9, 27, ...

Can you help her?

Solution

Observe that 3 1 = 9 3 = 27 9 =3 3 1 = 9 3 = 27 9 = 3

स्रोत : www.cuemath.com

## What is the 5th term in a geometric sequence in which the first term is 2 and the common ratio is 3?

Answer (1 of 5): The geometric sequence with first term 2 and common ratio 3 is? Listing; 2, 6, 18, 54, 162, 486, By inspection we find that the 5th term is 162.

What is the 5th term in a geometric sequence in which the first term is 2 and the common ratio is 3?

In the geometric sequence, a = 2 and r = 3,

The 5th term is thus ar^4 = 2*3^4 = 162.

Related questions

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What is the 5th term of a geometric sequence whose first term is 3 and the common ratio is 6?

What is the sexth term of a geometric sequence where the first term is 6 and the common ratio is -3?

Sally Barton 1y Related

What is the 5th term of a geometric sequence whose first term is 3 and the common ratio is 6?

If you understand what a geometric sequence is you can answer this yourself. The first term us easy to understand it is simply that - the first term ie the 1st term. You have a geometric sequence so this tells you that to get from one term to the next you always multiply by the same number. This number is called the common ratio (it gets that name because if you take any two successive terms in the sequence and divide the later one by the earlier one you will always get this number). So to get the second term you multiply by 6, multiplying by 6 again gives the 3rd term. Notice that for 1st, 2n

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Abhishek Mishra

Neuroscientist at Max Planck Institute of NeurobiologyUpdated 4y

In a G.P. you keep on multiplying the common ratio (r), as you proceed i.e. for 2nd term you multiply r to first term (a) one time making 2nd term equal to ar . If you proceed - the third term is ar^2, the fourth term is ar^3, fifth term is ar^4, nth term would be ar^(n-1).

The answer here would be 2 * 3^4 = 2*81 = 162

Watch my video for detailed explanation :

Nolan Quintal

Associate Science from Windward Community College (Expected 2024)Author has 2.1K answers and 873.5K answer views1y

The geometric sequence with first term 2 and common ratio 3 is?

Listing; 2, 6, 18, 54, 162, 486,

By inspection we find that the 5th term is 162.

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The first term of a geometric sequence is 8 and its common ratio is 3, how will you find the second term?

What is the first term of a geometric sequence with a common ratio of 2 and the third term is 32?

John K Williamsson

Let me show you how to use a calculator to find an answer to a similar question:

What is the fourth term in a geometric sequence in which the first term is 4 and the common ratio is 6?

To solve this question, I am using my Texas Instruments TI-84 PLUS CE Graphing Calculator

but you should be able to use almost any calculator:

Type your first term (4 in this example) and press ENTER

Press the times key then type your common ratio (6 in this example)

(to get the fourth term, press Enter three times)

Above, we see that our four terms are 4, 24, 144,

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R.N Singh

First term (a)=2 Common ratio (r)=3

nth term of geometric series Tn=ar^n-1

T5=2(3)5–1=2(3)4=2×81=162

Geoffrey Richard

Former K-12 Math TeacherAuthor has 67 answers and 10.5K answer viewsOct 8

Related

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the fifth term?

General formula for a geometric sequence is g(n) = g(1)*r^(n-1) where g(1) is the first term (3), r is the common ratio (2) and n is the number of the term you want to find (5)

So by substituting these values into the general formula you would have

g(5) = 3*2^(5–1) = 3*2^4 = 3*16 = 48

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