Sequence, Series and Arithmetic Mean

A sequence is a function whose domain is a subset of natural numbers, representing an ordered list of terms \( f_1, f_2, f_3, \ldots, f_n \) where \( f_n = f(n) \). A real sequence is one whose range is a subset of real numbers \( \mathbb{R} \).

A series is the sum of the terms of a sequence, expressed as \( a_1 + a_2 + a_3 + \ldots + a_n \). A progression is a sequence following a specific rule.

Series can be finite (with a finite number of terms) or infinite (with infinitely many terms).

Arithmetic Mean (A.M.)

When three quantities \( a, A, b \) are in arithmetic progression (A.P.), the middle term \( A \) is called the arithmetic mean between \( a \) and \( b \). This means:

\[ A - a = b - A \]

Adding \( A \) to both sides and rearranging, we get:

\[ 2A = a + b \implies A = \frac{a + b}{2} \]

Thus, the arithmetic mean between two numbers is half their sum.

n Arithmetic Means Between Two Numbers

Let \( A_1, A_2, \ldots, A_n \) be the \( n \) arithmetic means inserted between two numbers \( a \) and \( b \). The common difference \( d \) of the resulting A.P. is:

\[ d = \frac{b - a}{n + 1} \]

The arithmetic means are then:

\[ A_k = a + kd = a + k \times \frac{b - a}{n + 1}, \quad k = 1, 2, \ldots, n \]

Worked Example

Find 3 arithmetic means between 4 and 19.

Solution:

Number of means \( n = 3 \), \( a = 4 \), \( b = 19 \).

Calculate common difference:

\[ d = \frac{19 - 4}{3 + 1} = \frac{15}{4} = 3.75 \]

Arithmetic means:

• \( A_1 = 4 + 1 \times 3.75 = 7.75 \)
• \( A_2 = 4 + 2 \times 3.75 = 11.5 \)
• \( A_3 = 4 + 3 \times 3.75 = 15.25 \)

Practice Set

• Level 1 – Easy: Find the arithmetic mean between 10 and 20.
• Level 2 – Moderate: Insert 4 arithmetic means between 7 and 27.
• Level 3 – Challenging: Find 5 arithmetic means between \( -3 \) and 12, and verify the sum of these means.

Answer Key

• Level 1: \( \frac{10 + 20}{2} = 15 \)
• Level 2: \( d = \frac{27 - 7}{5} = 4 \), means: 11, 15, 19, 23
• Level 3: \( d = \frac{12 - (-3)}{6} = \frac{15}{6} = 2.5 \), means: \(-0.5, 2, 4.5, 7, 9.5\), sum = 22.5

Quick Reference

Concept	Formula
Arithmetic Mean between \( a \) and \( b \)	\( \frac{a + b}{2} \)
n Arithmetic Means between \( a \) and \( b \)	\( A_k = a + k \times \frac{b - a}{n + 1} \)

Glossary

• Sequence: Ordered list of numbers.
• Series: Sum of sequence terms.
• Arithmetic Progression (A.P.): Sequence with constant difference.
• Arithmetic Mean (A.M.): Middle term in A.P. between two numbers.

Geometric Progression and Sum to n Terms of Special Series

A geometric progression (G.P.) is a sequence where each term after the first is obtained by multiplying the previous term by a constant called the common ratio \( r \). Formally, for terms \( a_1, a_2, \ldots, a_n \),

\[ \frac{a_{k+1}}{a_k} = r, \quad \text{constant} \]

The sequence can be written as:

\[ a, ar, ar^2, ar^3, \ldots \]

General Term of a G.P.

The \( n^{th} \) term \( T_n \) of a G.P. with first term \( a \) and common ratio \( r \) is:

\[ T_n = a r^{n-1} \]

Term from the End

For a finite G.P. with \( n \) terms and last term \( l \), the \( m^{th} \) term from the end is the \( (n - m + 1)^{th} \) term from the beginning:

\[ T_{n - m + 1} = l \left( \frac{1}{r} \right)^{m - 1} \]

Sum of First n Terms of a G.P.

The sum \( S_n \) of the first \( n \) terms is given by:

For \( r \neq 1 \),

\[ S_n = \begin{cases} \frac{a(1 - r^n)}{1 - r}, & |r| < 1 \\ \frac{a(r^n - 1)}{r - 1}, & r > 1 \end{cases} \]

Infinite Sum of a G.P.

If \( |r| < 1 \), the infinite sum is:

\[ S_\infty = \frac{a}{1 - r} \]

Geometric Mean (G.M.)

The geometric mean between two positive numbers \( a \) and \( b \) is:

\[ G = \sqrt{ab} \]

If \( a, b, c \) are in G.P., then:

\[ b^2 = ac \]

Important Properties

• Reciprocals of terms in a G.P. form a G.P.
• If \( G_1, G_2, \ldots, G_n \) are inserted between \( a \) and \( b \) in G.P., then:

\[ G_k = a \left( \frac{b}{a} \right)^{\frac{k+1}{n+1}}, \quad 1 \leq k \leq n \]

• Product of terms equidistant from beginning and end equals \( a \times l \).
• Raising each term of a G.P. to a power preserves the G.P. property.
• For three terms in G.P., choose \( a/r, a, ar \).
• For four terms in G.P., choose \( a/r^3, a/r, ar, ar^3 \).
• For five terms in G.P., choose \( a/r^2, a/r, a, ar, ar^2 \).

Worked Example

Find the sum of the first 5 terms of a G.P. where \( a = 3 \) and \( r = 2 \).

Solution:

Using the formula for sum of first \( n \) terms:

\[ S_5 = \frac{3(2^5 - 1)}{2 - 1} = \frac{3(32 - 1)}{1} = 3 \times 31 = 93 \]

Practice Set

• Level 1 – Easy: Find the 4th term of the G.P. 2, 6, 18, ...
• Level 2 – Moderate: Find the sum of first 6 terms of a G.P. with \( a = 5 \) and \( r = 3 \).
• Level 3 – Challenging: Insert 3 geometric means between 2 and 162.

Answer Key

• Level 1: \( T_4 = 2 \times 3^{3} = 54 \)
• Level 2: \( S_6 = \frac{5(3^6 - 1)}{3 - 1} = \frac{5(729 - 1)}{2} = \frac{5 \times 728}{2} = 1820 \)
• Level 3: Common ratio \( r = \sqrt[4]{\frac{162}{2}} = 3 \), means: 6, 18, 54

Quick Reference

Concept	Formula
n-th term of G.P.	\( T_n = a r^{n-1} \)
Sum of first n terms (\( r \neq 1 \))	\( S_n = \frac{a(1 - r^n)}{1 - r} \) if \( |r| < 1 \), else \( S_n = \frac{a(r^n - 1)}{r - 1} \)
Sum to infinity (\( |r| < 1 \))	\( S_\infty = \frac{a}{1 - r} \)
Geometric mean between \( a \) and \( b \)	\( \sqrt{ab} \)

Glossary

• Geometric Progression (G.P.): Sequence with constant ratio.
• Common Ratio: Constant multiplier between terms in G.P.
• Geometric Mean (G.M.): Square root of product of two numbers.
• Sum to n terms: Total of first \( n \) terms of a series.