Arithmetic and Geometric Sequences

Arithmetic and Geometric Sequences are two fundamental types of number patterns in mathematics. An Arithmetic Sequence has a constant difference between consecutive terms, while a Geometric Sequence has a constant ratio between terms. Both are widely used to solve problems involving series, progressions, and real-life applications. They also have specific formulas to calculate the nth term and the sum of terms. Understanding these sequences is essential for tackling questions in algebra, calculus, and competitive exams like JEE.

1.0What is an Arithmetic Sequence?

An Arithmetic Sequence (also called an Arithmetic Progression or AP) is a sequence of numbers where each term after the first is obtained by adding a constant value called the common difference (d).

• General Form: $a,a+d,a+2d,a+3d...$
• Formula for nth Term of an Arithmetic Sequence: $a_n=a+(n-1)d$
• Sum of n Terms (Arithmetic Series):$S_n=\frac{n}{2}[2a+(n-1)d]$

2.0What is a Geometric Sequence?

A Geometric Sequence (also called a Geometric Progression or GP) is a sequence of numbers where each term after the first is obtained by multiplying by a constant called the common ratio (r).

• General Form: $a,ar,a{r}^2,a{r}^3,....$
• Formula for nth Term of a Geometric Sequence: $a_n=a{r}^{n-1}$
• Sum of n Terms (Geometric Series): For $r\ne 1$:  $S_n=a\frac{r^n-1}{r-1}$

3.0Difference Between Arithmetic and Geometric Sequences

4.0Arithmetic and Geometric Sequences and Series Formulas

Here’s a quick recap of the most important arithmetic and geometric sequences and series formulas:

Arithmetic Sequence Formulas:

• nth Term: $a_n=a+(n-1)d$
• Sum of n Terms: $S_n=\frac{n}{2}[2a+(n-1)d]$

Geometric Sequence Formulas:

• nth Term: $a_n=a{r}^{n-1}$
• Sum of n Terms: $S_n=a\frac{r^n-1}{r-1}$ ($forr\ne 1$)

5.0How to Tell if a Sequence is Arithmetic or Geometric?

• Arithmetic: See if the difference between consecutive terms remains the same.
• Geometric: See if the ratio between consecutive terms stays the same.

Example 1:

Sequence: 2, 4, 6, 8, ...

Difference between terms: 2 → Arithmetic.

Example 2:

Sequence: 3, 6, 12, 24, ...

Ratio between terms: 2 → Geometric.

6.0Similarities Between Arithmetic and Geometric Sequences

1. Both follow a specific pattern/rule.
2. Both have formulas for the nth term and the sum of terms.
3. Both are commonly used in solving real-life problems related to finance, physics, and more.
4. Both are classified as sequences and series in mathematics.

7.0Arithmetic and Geometric Sequences Problems

Problem 1: Find the 10th term of the AP: 5, 9, 13, ….

Solution:
a = 5,  d = 4

$a_{10}=5+(10-1).4=5+36=41$

Problem 2: Find the 5th term of the GP: 3, 6, 12, … .

Solution:
a = 3,  r = 2

$a_5=3.2^{5-1}=3.16=48$

Problem 3: Identify whether the following sequence is arithmetic or geometric:

10, 20, 40, 80, … 

Solution:
Ratio between terms: $\frac{20}{10}=2,\frac{40}{20}=2$→ Geometric Sequence.

8.0Solved Examples on Arithmetic and Geometric Sequences (JEE & Advanced Level)

Example 1: Calculate the infinite sum of the series.:$3+6x+12x^2+24x^3+....$, where $|x|<\frac{1}{2}$.

Solution:
Here,
a = 3,

r = 2x,

|r| < 1 (since $|x|<\frac{1}{2}$)

Sum to infinity: $S_{\infty }=\frac{a}{1-r}=\frac{3}{1-2x}$

Answer: $\frac{3}{1-2x}$

Example 2: Evaluate the sum of the infinite series:$\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+...$

Solution:

This is an arithmetic-geometric series where:

• Arithmetic part: $1,2,3,...$
• Geometric part: $\frac{1}{2},\frac{1}{4},\frac{1}{8},....$

Common ratio $r=\frac{1}{2}$.

Sum formula for infinite arithmetic-geometric series: $S=\frac{a}{1-r}+\frac{d.r}{(1-r{)}^2}$

Here,

$a=\frac{1}{2},d=\frac{1}{2},r=\frac{1}{2}$

$S=\frac{\frac{1}{2}}{1-\frac{1}{2}}+\frac{\frac{1}{2}.\frac{1}{2}}{(1-\frac{1}{2}{)}^2}$

$S=\frac{\frac{1}{2}}{\frac{1}{2}}+\frac{\frac{1}{4}}{\frac{1}{4}}$ = 1+1=2

Answer: 2

Example 3: Evaluate ${\sum }_{n=1}^{\infty }n.(\frac{1}{3}{)}^n$.

Solution:

Here, $a=\frac{1}{3},r=\frac{1}{3},d=\frac{1}{3}$

Using sum of infinite arithmetic-geometric series:

$S=\frac{a}{(1-r{)}^2}$

$S=\frac{\frac{1}{3}}{(1-\frac{1}{3}{)}^2}=\frac{\frac{1}{3}}{(\frac{2}{3}{)}^2}=\frac{\frac{1}{3}}{\frac{4}{9}}=\frac{9}{12}=\frac{3}{4}$

Answer: $\frac{3}{4}$

Example 4: Evaluate the sum of the infinite series: $1+\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+....$

Solution:

This is an Arithmetic-Geometric Series:

• Arithmetic part: 1,3,5,7,.... (odd numbers; a = 1, d = 2)
• Geometric part: $1,\frac{1}{2},\frac{1}{4},\frac{1}{8},...$ with $r=\frac{1}{2}$.

Use the infinite sum formula: $S_{\infty }=\frac{a}{1-r}+\frac{d.r}{(1-r{)}^2}$

Here, $a=1,d=2,r=\frac{1}{2}$

$S_{\infty }=\frac{1}{1-\frac{1}{2}}+\frac{2.\frac{1}{2}}{(1-\frac{1}{2}{)}^2}$

$S_{\infty }=\frac{1}{\frac{1}{2}}+\frac{1}{(\frac{1}{2}{)}^2}$

$S_{\infty }=2+4=6$

Answer: 6

Example 5: Find the sum of the infinite series: $2.\frac{1}{3}+4.(\frac{1}{3}{)}^3+6.(\frac{1}{3}{)}^3+....$

Solution:

This is an Arithmetic-Geometric Series:

• Arithmetic part: 2,4,6,... with a = 2, d = 2.
• Geometric part: $\frac{1}{3},\frac{1}{9},\frac{1}{27},...$with $r=\frac{1}{3}$.

Sum to infinity formula: $S_{\infty }=\frac{a}{1-r}+\frac{d.r}{(1-r{)}^2}$

Substitute values:

$S_{\infty }=21-13+2\cdot 13(1-13)2$

$S_{\infty }=\frac{2}{1-\frac{1}{3}}+\frac{2.\frac{1}{3}}{(1-\frac{1}{3}{)}^2}$

$S_{\infty }=\frac{2}{\frac{2}{3}}+\frac{\frac{2}{3}}{(\frac{2}{3}{)}^2}$

$S_{\infty }=3+\frac{\frac{2}{3}}{\frac{4}{9}}=3+\frac{2}{3}.\frac{9}{4}$

$S_{\infty }=3+\frac{3}{2}=\frac{9}{2}$

Answer: $\frac{9}{2}$

Example 6: Evaluate the infinite sum: $5.\frac{1}{2}+10(\frac{1}{2}{)}^2+15(\frac{1}{2}{)}^3+...$

Solution:

Arithmetic part: 5,10,15,... with a = 5, d = 5.

Geometric part: $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$with $r=\frac{1}{2}$.

Sum to infinity: $S_{\infty }=\frac{a}{1-r}+\frac{dr}{(1-r{)}^2}$

Substitute:

$S_{\infty }=\frac{5}{1-\frac{1}{2}}+\frac{5.\frac{1}{2}}{(1-\frac{1}{2}{)}^2}$

$S_{\infty }=\frac{5}{\frac{1}{2}}+\frac{\frac{5}{2}}{(\frac{1}{2}{)}^2}$

$S_{\infty }=10+\frac{\frac{5}{2}}{\frac{1}{4}}$

$S_{\infty }=10+10=20$

Answer: 20

Example 7: Find ${\sum }_{n=1}^{\infty }{n}^2.r^n$for |r| < 1.

Solution:

We use calculus tricks (advanced JEE method):

We know,

${\sum }_{n=1}^{\infty }n{r}^n=\frac{r}{(1-r{)}^2}$

Differentiate both sides w.r.t. r:

${\sum }_{n=1}^{\infty }{n}^2.r^{n-1}=\frac{1+r}{(1-r{)}^3}$

Multiply both sides by r:

${\sum }_{n=1}^{\infty }{n}^2{r}^n=\frac{r(1+r)}{(1-r{)}^3}$

Answer: $\frac{r(1+r)}{(1-r{)}^3}$ (Sum for |r| < 1)

9.0Practice Questions on Arithmetic and Geometric Sequences

Question 1: Find the sum to infinity of $4+8x+12x^2+16x^3+...$ for $|x|<\frac{1}{2}$.

Question 2: Evaluate ${\sum }_{n=1}^{\infty }n.(\frac{1}{4}{)}^n$.

Question 3: Find the sum of the infinite series $\frac{2}{3}+\frac{4}{9}+\frac{6}{27}+....$:  

Question 4: Find the sum to infinity of $\frac{1}{5}+\frac{2}{25}+\frac{3}{125}+...$

Question 5: A series has terms of the form $n.r^n$ with |r| < 1. Derive the general formula for its sum to infinity.

10.0Sample Questions on Arithmetic and Geometric Sequences

Q1. What is an Arithmetic-Geometric Sequence?

Ans: An Arithmetic-Geometric Sequence is a sequence where terms are a product of an arithmetic sequence and a geometric sequence, generally of the form:

$a.r^n+(a+d).r^{n+1}+(a+2d).r^{n+2}+...$

Q2. What is the formula for sum to infinity of an arithmetic-geometric series?

Ans: For |r| < 1: $S_n=\frac{a}{1-r}+\frac{d.r}{(1-r{)}^2}$