A sequence is a collection of objects (or events) arranged logically in mathematics. A sum of a sequence of terms is referred to as a series. In other words, a series is a collection of numbers connected by addition operations. Many types of sequences (also called progressions) and series are studied in mathematics such as arithmetic progression, geometric progression, harmonic progression, etc.

Let’s understand what is geometric progression and what the different formulas used to solve problems are based on it.

## What is a Geometric Progression?

A geometric progression(abbreviated as GP) is a progression where the successive terms bear a constant ratio known as a common ratio.

It is generally represented in form $a$, $ar$, $ar^2$, … where $a$ is the first term and $r$ is the common ratio of the progression. The common ratio can have both negative as well as positive values.

**Note:** To get the common ratio $r$ of a geometric progression we divide any succeeding term by its preceding term, i.e., $r = \frac{a_n}{a_{n-1}}$.

For example, $1, 2, 4, 8, 16, 32$, … is a geometric progression as the ratio between every two consecutive terms is the same (i.e., $2$). $\frac{2}{1} = \frac{4}{2} = \frac{8}{4} = \frac{16}{8} = \frac{32}{16} = … = 2$. We can also notice that every term (except the first term) of this GP is obtained by multiplying $2$ by its previous term.

Thus, a geometric progression, in general, can be written as $a$, $ar$, $ar^2$, $ar^3$, …, where $a$ is a first term and $r$ is a common ratio.

### Examples of Geometric Progression

**Example 1:** The amount of air present in a cylinder when a vacuum pump removes $\frac{1}{4}$ of the air remaining in the cylinder at a time.

Let’s consider that the initial amount of air in a cylinder = $100$ cubic units

Amount of air removed by vacuum pump after first time = $\frac{1}{4} \times 100 = 25$. Amount of air left in the cylinder = $100 – 25 = 75$ cubic units.

Amount of air removed by vacuum pump after second time = $\frac{1}{4} \times 75 = 18.75$. Amount of air left in the cylinder = $75 – 18.75 = 56.25$ cubic units.

Amount of air removed by vacuum pump after third time = $\frac{1}{4} \times 56.25 = 14.0625$. Amount of air left in the cylinder = $56.25 – 14.0625 = 42.1875$ cubic units.

The amount of air in the cylinder(in cubic units) after each use of the vacuum pump is $100$, $75$, $56.25$, $42.1875$, … which is not an AP.

Ratio between $2^{nd}$ and $1^{st}$ terms is $\frac{75}{100} = \frac{3}{4}$

Ratio between $3^{rd}$ and $2^{nd}$ terms is $\frac{56.25}{75} = \frac{3}{4}$

Ratio between $4^{th}$ and $3^{rd}$ terms is $\frac{42.1875}{56.25} = \frac{3}{4}$

The ratio between consecutive terms of the sequence is the same.

**Example 2:** The number of bacteria in a certain culture doubles every hour. If there were $30$ bacteria present in the culture originally, then the number of bacteria in the culture after every hour.

Number of bacteria in $1^{st}$ hour = $30$.

Number of bacteria in $2^{nd}$ hour = $60$.

Number of bacteria in $3^{rd}$ hour = $120$.

Number of bacteria in $4^{th}$ hour = $240$.

Ratio between $2^{nd}$ and $1^{st}$ terms is $\frac{60}{30} = 2$

Ratio between $3^{rd}$ and $2^{nd}$ terms is $\frac{120}{60} = 2$

Ratio between $4^{th}$ and $3^{rd}$ terms is $\frac{240}{120} = 2$

The ratio between consecutive terms of the sequence is the same.

**Example 3:** What will ₹$500$ amount to in $10$ years after its deposit in a bank which pays an annual interest rate of $10\%$ compounded annually?

Amount of money in the beginning of $1^{st}$ year = ₹$500$

Amount of money in the beginning of $2^{st}$ year = $500\left( 1 + \frac{10}{100}\right) =$ ₹$ 550$

Amount of money in the beginning of $3^{rd}$ year = $550\left( 1 + \frac{10}{100}\right) =$ ₹$ 605$

Amount of money in the beginning of $4^{th}$ year = $605\left( 1 + \frac{10}{100}\right) =$ ₹$ 665.50$

Ratio between $2^{nd}$ and $1^{st}$ terms is $\frac{550}{500} = \frac{11}{10}$

Ratio between $3^{rd}$ and $2^{nd}$ terms is $\frac{605}{550} = \frac{11}{10}$

Ratio between $4^{th}$ and $3^{rd}$ terms is $\frac{665.50}{605} = \frac{11}{10}$

The ratio between consecutive terms of the sequence is the same.

**Note: **

- A GP is an increasing GP when the common ratio $r \gt 1$
- A GP is a decreasing GP when the common ratio $r \lt 1$
- A GP is a constant GP when the common ratio $r = 1$

## nth Term of Geometric Progression

The general term (or) $n^{th}$ term of a GP whose first term is $a$ and the common ratio is $r$ is given by the formula $a_n = ar^{n-1}$.

For example, to find the general term (or $n^{th}$) term of the progression $3, 9, 27, 81, 243,$ … we substitute the first term, $a = 3$, and the common ratio, $r = 3$ in the formula for the $n^{th}$ term formula.

Using the formula we get, $a_n = ar^{n – 1} = 3 \times 3^{n – 1} = 3^n$. Thus, the general term (or) $n^{th}$ term of this GP is $a_n = 3^n$.

### Examples on General Term of Geometric Progression

**Example 1:** Find the general term of a GP whose first term is $\frac{5}{2}$ and the common ratio is $\frac{1}{2}$.

The first term $a = \frac{5}{2}$ and the common ratio $r = \frac{1}{2}$.

Using the formula we get, $a_n = ar^{n – 1} = \frac{5}{2} \times \left(\frac{1}{2} \right)^{n – 1} = \frac{5^{n – 1}}{2^n}$

**Example 2:** Find the general term of a GP whose first term is $x^3$ and the common ratio is $x^2$.

The first term $a = x^3$ and the common ratio $r = x^2$.

Using the formula we get, $a_n = ar^{n – 1} = x^3 \times \left(x^2 \right)^{n – 1} = x^3 \times x^{2n – 2} = x^{3 + 2n – 2} = x^{2n + 1}$.

**Example 3:** Find the general term of a GP whose first term is $1$ and the common ratio is $-a$.

The first term $a = 1$ and the common ratio $r = -a$.

Using the formula we get, $a_n = ar^{n – 1} = 1 \times (-a)^{n – 1} = (-1)^{n – 1}a^{n – 1}$.

## Derivation of the nth Term of Geometric Progression

Let $a_1$, $a_2$, $a_3$, . . . be an GP whose first term $a_1$ is $a$ and the common difference is $r$.

Then, the second term $a_2 = ar = ar^{2 – 1}$

The third term $a_3 = a_2 \times r = ar \times r = ar^2 = ar^{3 – 1}$

The fourth term $a_4 = a_3 \times r = ar^2 \times r = ar^3 = ar^{4 – 1}$

Looking at the pattern, we can say that the $n^{th}$ term $a_n = ar^{n – 1}$.

So, the $n^{th}$ term an of the GP with first term $a$ and common ratio $r$ is given by $a_n = ar^{n – 1}$.

### Examples on the nth Term of Geometric Progression

**Example 1:** Find the $11^{th}$ term of a GP $1, 3, 9, 27, …$.

The first term $a = 1$

The common ratio $r = \frac{3}{1} = 3$

$n = 11$

Formula for $n^{th}$ term an of the GP is $a_n = ar^{n – 1}$.

Substituting the values we get $a_n = ar^{n – 1} = 1 \times 3^{11 – 1} = 3^10 = 59049$

Therefore, the $11^{th}$ term of a GP $1, 3, 9, 27, …$ is $59049$.

**Example 2:** Find the $20^{th}$ and $n^{th}$ terms of the G.P. $\frac{5}{2}$, $\frac{5}{4}$, $\frac{5}{8}$, ….

The first term $a = \frac{5}{2}$

The common ratio $r = \frac{\frac{5}{4}}{\frac{5}{2}} = \frac{1}{2}$

$n = 20$

Formula for $n^{th}$ term an of the GP is $a_n = ar^{n – 1}$.

Substituting the values we get $a_n = ar^{n – 1} = \frac{5}{2} \times \left(\frac{1}{2} \right)^{20 – 1} = \frac{5}{2} \times \left( \frac{1}{2}\right)^{19} = \frac{5}{2^{20}} = \frac{5}{1048576}$

And the $n^{th}$ term is $\frac{5}{2} \times \left(\frac{1}{2} \right)^{n – 1} = \frac{5}{2^n}$.

**Example 3:** The $5^{th}$, $8^{th}$, and $11^{th}$ terms of a G.P. are $p$, $q$ and $s$, respectively. Show that $q^2 = ps$.

Let the first term and the common ratio of the GP be $a$ and $r$ respectively.

Therefore, $a \times r^{5 – 1} = p => ar^4 = p$

$a \times r^{8 – 1} = p => ar^7 = q$

And $a \times r^{11 – 1} = p => ar^{10} = s$

Therefore, $ps = ar^4 \times ar{10} = a^2r^{14} = \left(ar^7 \right)^2 = q^2$

**Example 4:** The $4^{th}$ term of a G.P. is the square of its second term, and the first term is $-3$. Determine its $7^{th} term.

The first term of the G.P. $a = -3$.

Let the common ratio = $r$

The $4^{th}$ term = $-3 \times r^{4 – 1} = -3r^3$ and the second term = $-3 \times r^{2 – 1} = -3r$

The $4^{th}$ term of a G.P. is the square of its second term, therefore, $-3r^3 = -3r => -3r^3 + 3r = 0 => -3r\left(r^2 – 1 \right) = 0 => r = 0, r = 1, \text{ or } r = -1$

$r$ cannot be zero, therefore, $r = 1$ or $r = -1$

$7^{th}$ term of the GP is $ar^{7 – 1} = ar^6$

When $r = 1$, the $7^{th}$ term is $ar^6 = -3 \times 1^6 = -3$

When $r = -1$, the $7^{th}$ term is $ar^6 = -3 \times (-1)^6 = -3$

Therefore, the $7^{th}$ term is $-3$.

## Sum of n Terms of a Geometric Progression

The formula to find the sum of the first $n$ terms of a GP is

- $\text{S}_n = \frac{a\left(1 – r^n \right)}{1 – r}$, when $r \lt 1$
- $\text{S}_n = \frac{a\left(r^n – 1 \right)}{r – 1}$, when $r \gt 1$

For example, to find the sum of upto $15^{th}$ term of the progression $1, 2, 4, 8, 16$ … we substitute the first term, $a = 1$, the common ratio, $r = 2$ and $n = 15$ in the sum formula.

Since, $r \gt 1$, using the formula we get, $\text{S}_{15} = \frac{1\left(2^{15} – 1 \right)}{2 – 1} = \frac{1 \left(32768 – 1 \right)}{1} = 32768 – 1 = 32767$

### Examples on Sum of n Terms of a Geometric Progression

**Example 1:** Find the sum of the first $n$ terms and the sum of the first $5$ terms of the geometric series $1$, $\frac{2}{3}$, $\frac{4}{9}$, …

For the given GP, $a = 1$, $r = \frac{2}{3}$

Since, $r \lt 1$, therefore, $\text{S}_n = \frac{a\left(1 – r^n \right)}{1 – r}$

Sum of first $n$ terms = $\text{S}_n = \frac{1\left(1 – \left(\frac{2}{3} \right)^n \right)}{1 – \frac{2}{3}} = \frac{\frac{3^n – 2^n}{ 3^n}}{\frac{1}{3}} = \frac{3^n – 2^n}{3^n} \times \frac{3}{1} = \frac{3^n – 2^n}{3^{n – 1}}$

Now, substitute $n = 5$ to get the sum of the first $5$ terms of the GP.

Sum of the first $5$ terms of the GP is $\frac{3^5 – 2^5}{3^{5 – 1}} = \frac{243 – 32}{3^4} = \frac{211}{81}$

**Example 2:** How many terms of the GP $3$, $\frac{3}{2}$, $\frac{3}{4}$, $\frac{3}{8}$, … are needed to give the sum $\frac{3069}{512}$?

The given GP is $3$, $\frac{3}{2}$, $\frac{3}{4}$, $\frac{3}{8}$, …

The first term $a = 3$ and the common ratio = $\frac{\frac{3}{2}}{3} = \frac{1}{2}$.

Let the number of terms = $n$, therefore $\text{S}_n = \frac{3069}{512}$.

Since $r \lt 1$, therefore, $\text{S}_n = \frac{a\left(1 – r^n \right)}{1 – r}$.

$=> \frac{3069}{512} = \frac{3 \times \left(1 – \left( \frac{1}{2} \right)^n \right)}{1 – \frac{1}{2}}$

$=> \frac{3069}{512} = \frac{3 \times \left(1 – \left( \frac{1}{2} \right)^n \right)}{\frac{1}{2}}$

$=> \frac{3069}{512} = \frac{3}{2} \times \left(1 – \frac{1}{2} \right)^n$

$=> \frac{3069}{512} \times \frac{2}{3} = \left(1 – \frac{1}{2} \right)^n$

$=> \frac{1023}{256} = \left(1 – \frac{1}{2} \right)^n$

$=> \frac{1023}{256} = \left(\frac{1}{2} \right)^n => n = 8$

Therefore $8$ terms of the GP $3$, $\frac{3}{2}$, $\frac{3}{4}$, $\frac{3}{8}$, … are needed to give the sum $\frac{3069}{512}$.

## Key Takeaways

- In a geometric progression, each successive term is obtained by multiplying the common ratio by its preceding term.
- The formula for the $n^{th}$ term of a geometric progression whose first term is $a$ and the common ratio is $r$ is $a_n = ar^{n-1}$.
- The sum of $n$ terms in GP whose first term is $a$ and the common ratio is $r$ can be calculated using the formula: $\text{S}_n = \frac{a\left(1 – r^n \right)}{1 – r}$.
- The sum of the infinite GP formula is given as $\text{S}_n = \frac{a}{1 – r}$ where $|r| \lt 1$.

## Practice Problems

- What is a geometric progression?
- Check whether the following sequence of numbers forms a geometric progression.
- $4, 8, 12, 16, …$
- $1, 4, 16, 64, …$
- $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, …$

- Find the general term of the following GPs
- $1, 2, 4, 8, 16, …$
- $1, 0.1, 0.01, 0.001, 0.0001, …$
- $\frac{2}{3}, \frac{1}{3}, \frac{1}{6}$

- Find the sum of the sequence $7, 77, 777, 7777, …$ to $n$ terms.
- For what values of x, the numbers $-\frac{2}{7}, x, \frac{2}{7}$ are in GP?
- Find the sum upto $10$ terms of the GP $\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, …$

## FAQs

### What is a geometric progression?

A geometric progression(abbreviated as GP) is a progression where the successive terms bear a constant ratio known as a common ratio.

It is generally represented in form $a$, $ar$, $ar^2$, … where $a$ is the first term and $r$ is the common ratio of the progression. The common ratio can have both negative as well as positive values.

### What are GP formulas?

The important GP formulas for a geometric progression with the first term $a$ and the common ratio $r$ are

a) $n^{th}$ term, $a_n = ar^{n-1}$

b) Sum of the first $n$ terms, $\text{S}_n = \frac{a\left(1 – r^n \right)}{1 – r}$ when $r \ne 1$. When $r = 1$, $\text{S}_n = na$

c) Sum of infinite terms (when $|r| \lt 1$), $\text{S}_n = \frac{a}{1 – r}$ where $|r| \lt 1$

### What is r in GP formula?

In geometric progression, $r$ is the common ratio of the two consecutive terms. The common ratio can have both negative as well as positive values. In order to get the next term in the GP, we have to multiply with a fixed term known as the common ratio, every time, and if we want to find the preceding term in the progression, we just have to divide the term by the same common ratio.

### How to find the common ratio in geometric progression?

The common ratio is calculated by finding the ratio of any term to its preceding term.

For example, in the G.P. $1, 3, 9, …$, the common ratio is $r = \frac{3}{1} = 3$.

### What is the difference between arithmetic progression and geometric progression?

If each successive term of a progression is less than the preceding term by a fixed number, then the progression is an arithmetic progression (AP). If each successive term of a progression is a product of the preceding term and a fixed number, then the progression is a geometric progression. The ratio of two terms in an AP is not the same throughout but in GP, it is the same throughout.

## Conclusion

A geometric progression(abbreviated as GP) is a progression where the successive terms bear a constant ratio known as a common ratio. It is generally represented in form $a$, $ar$, $ar^2$, … where $a$ is the first term and $r$ is the common ratio of the progression. The common ratio can have both negative as well as positive values.