Important Formulas Arithmetic Progression and Geometric Progression - Quantitative

Basic Concept on Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive members is always the same. This fixed difference is called the common difference.

• First term is denoted by a.
• Common difference is denoted by d.
• nth term is denoted by t_n or T_n.
• Sum of first n terms is denoted by S_n.
• Example: 4, 8, 12, 16, ... is an AP with a = 4 and d = 4.

Formula of Arithmetic Progression

nth term of an AP

• Formula: T_n = a + (n - 1)d
• where T_n = nth term, a = first term, d = common difference, and n = number of terms.

Number of terms in an AP

If the first term is a, the last term is l, and the common difference is d, then the number of terms n satisfies the relation

l = a + (n - 1)d

Thus,

n =  (l - a)/d + 1

Sum of first n terms in an AP

The sum of the first n terms, S_n, can be obtained in two common forms.

S_n = n/2 [2a + (n - 1)d]

or, if the last term l = T_n is known,

S_n = n/2 (a + l)

where a = first term, d = common difference, and T_n = l = a + (n - 1)d.

Arithmetic Mean

If three numbers a, b, c are in AP, then b is the arithmetic mean of a and c. In symbols,

b = (a + c)/2

Some other important formulas of Arithmetic Progression

Sum of first n natural numbers

The sum of the first n natural numbers is given by S = n(n + 1)/2. A standard derivation is shown below.

Write the sum forwards and backwards and add corresponding terms.

Let S = 1 + 2 + 3 + ... + (n - 1) + n

Write S in reverse order: n + (n - 1) + (n - 2) + ... + 2 + 1

Add the two expressions termwise to obtain 2S = (n + 1) + (n + 1) + ... + (n + 1) (n terms)

Therefore 2S = n(n + 1)

S = n(n + 1)/2

Sum of squares of first n natural numbers

The sum of squares of the first n natural numbers is

S = 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6

Sum of first n odd numbers

The sum of the first n odd numbers is

S = n²

That is, 1 + 3 + 5 + ... (n terms) = n².

Sum of first n even numbers

The sum of the first n even numbers is

S = n(n + 1)

That is, 2 + 4 + 6 + ... (n terms) = n(n + 1).

Formulas for Geometric Progression (GP)

Common ratio

A geometric progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio.

If successive terms are a₁, a₂, a₃, ..., then the common ratio r is

r = a₂ / a₁ = a₃ / a₂ = ...

nth term of a GP

• Formula: a_n = a₁ · r^n-1
• where a₁ = first term, r = common ratio, and n = term index.

Sum of first n terms in a GP

For r ≠ 1, the sum of the first n terms is

S_n = a₁ (1 - rⁿ)/(1 - r)

Note: the algebraic form is the same for r > 1 and for |r| < 1; the displayed placeholders correspond to the original presentation. When r = 1, every term equals a₁ and S_n = n·a₁.

Sum of an infinite GP

If |r| < 1, an infinite GP has a finite sum. The sum to infinity is

S∞ = a₁ / (1 - r)

If |r| ≥ 1, the sum to infinity does not exist (it diverges).

Geometric Mean (GM)

• If two non-zero numbers a and b are in GP, then their geometric mean is GM = √(ab).
• If three non-zero numbers a, b, c are in GP, then their geometric mean is GM = (abc)^1/3.

Worked examples and short applications

Example 1: Find the 10th term of the AP 4, 8, 12, ...

Solution:

First term a = 4

Common difference d = 4

Use T_n = a + (n - 1)d

T₁₀ = 4 + (10 - 1) × 4

T₁₀ = 4 + 9 × 4

T₁₀ = 4 + 36

T₁₀ = 40

Example 2: Sum of first 20 terms of the AP 3, 7, 11, ...

Solution:

First term a = 3

Common difference d = 4

Number of terms n = 20

Use S_n = n/2 [2a + (n - 1)d]

S₂₀ = 20/2 [2 × 3 + (20 - 1) × 4]

S₂₀ = 10 [6 + 19 × 4]

S₂₀ = 10 [6 + 76]

S₂₀ = 10 × 82

S₂₀ = 820

Example 3: Sum to infinity of GP 5, 5/2, 5/4, ...

Solution:

First term a₁ = 5

Common ratio r = (5/2)/5 = 1/2

Since |r| < 1, S∞ = a₁/(1 - r)

S∞ = 5/(1 - 1/2)

S∞ = 5/(1/2)

S∞ = 10