8.16.a Arithmetic Progression - Concepts and Formulas

1. Definition of Arithmetic Progression

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is called the common difference.

General form of an AP:

a, a + d, a + 2d, a + 3d, a + 4d, ...

Where:

• a = first term
• d = common difference
• Each term is obtained by adding d to the previous term

Examples of AP:

Sequence	First Term (a)	Common Difference (d)
2, 5, 8, 11, 14, ...	2	3
10, 7, 4, 1, -2, ...	10	-3
3, 3, 3, 3, 3, ...	3	0
-5, -1, 3, 7, 11, ...	-5	4
0.5, 1.0, 1.5, 2.0, ...	0.5	0.5

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2. Common Difference

The common difference d is obtained by subtracting any term from its succeeding term:

d = a(n) - a(n-1)

For any three consecutive terms:

d = second term - first term = third term - second term

Key observations:

• If d > 0, the AP is increasing
• If d < 0, the AP is decreasing
• If d = 0, the AP is a constant sequence

Example:

Sequence: 7, 11, 15, 19, 23

d = 11 - 7 = 4
d = 15 - 11 = 4
d = 19 - 15 = 4
d = 23 - 19 = 4

Common difference = 4

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3. nth Term of an AP (General Term)

The nth term (also called the general term) of an AP is given by:

a(n) = a + (n - 1) * d

Where:

• a(n) = nth term
• a = first term
• n = position of the term
• d = common difference

Derivation:

1st term:  a(1) = a = a + (1-1)d = a + 0d
2nd term:  a(2) = a + d = a + (2-1)d = a + 1d
3rd term:  a(3) = a + 2d = a + (3-1)d = a + 2d
4th term:  a(4) = a + 3d = a + (4-1)d = a + 3d
...
nth term:  a(n) = a + (n-1)d

Example 1: Find the 15th term of AP: 3, 7, 11, 15, ...

a = 3, d = 7 - 3 = 4, n = 15

a(15) = 3 + (15 - 1) * 4
      = 3 + 14 * 4
      = 3 + 56
      = 59

Example 2: Find the 25th term of AP: 50, 45, 40, 35, ...

a = 50, d = 45 - 50 = -5, n = 25

a(25) = 50 + (25 - 1) * (-5)
      = 50 + 24 * (-5)
      = 50 - 120
      = -70

Finding position of a term:

If you know a term value and need to find its position:

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

Example: In the AP 5, 9, 13, 17, ..., which term is 101?

a = 5, d = 4, a(n) = 101

101 = 5 + (n - 1) * 4
96 = (n - 1) * 4
n - 1 = 24
n = 25

So, 101 is the 25th term.

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4. Sum of First n Terms of an AP

Formula 1: When first term (a) and common difference (d) are known

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

Formula 2: When first term (a) and last term (l) are known

S(n) = n/2 * (a + l)

Where:

• S(n) = sum of first n terms
• n = number of terms
• a = first term
• d = common difference
• l = last term = a + (n-1)d

Derivation of Formula 1:

S(n) = a + (a+d) + (a+2d) + ... + [a+(n-1)d]       ... (i)

Writing in reverse:
S(n) = [a+(n-1)d] + [a+(n-2)d] + ... + (a+d) + a   ... (ii)

Adding (i) and (ii):
2 * S(n) = [2a+(n-1)d] + [2a+(n-1)d] + ... (n times)
2 * S(n) = n * [2a + (n-1)d]
S(n) = n/2 * [2a + (n-1)d]

Example 1: Find the sum of first 20 terms of AP: 2, 5, 8, 11, ...

a = 2, d = 3, n = 20

S(20) = 20/2 * [2(2) + (20-1)(3)]
      = 10 * [4 + 57]
      = 10 * 61
      = 610

Example 2: Find the sum of AP: 5, 10, 15, ..., 200

a = 5, l = 200, d = 5

First find n:
200 = 5 + (n-1) * 5
195 = (n-1) * 5
n - 1 = 39
n = 40

S(40) = 40/2 * (5 + 200)
      = 20 * 205
      = 4100

Important Sum Formulas

Sum	Formula	Result
Sum of first n natural numbers	n(n+1)/2	1+2+3+...+n
Sum of first n even numbers	n(n+1)	2+4+6+...+2n
Sum of first n odd numbers	n^2	1+3+5+...+(2n-1)
Sum of squares of first n numbers	n(n+1)(2n+1)/6	1^2+2^2+...+n^2
Sum of cubes of first n numbers	[n(n+1)/2]^2	1^3+2^3+...+n^3

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5. Relationship Between S(n) and a(n)

The nth term can be obtained from the sum formula:

a(n) = S(n) - S(n-1)    for n >= 2
a(1) = S(1)

Example: If S(n) = 3n^2 + 5n, find the AP.

a(1) = S(1) = 3(1) + 5(1) = 8

a(2) = S(2) - S(1) = [3(4) + 5(2)] - 8 = 22 - 8 = 14

a(3) = S(3) - S(2) = [3(9) + 5(3)] - 22 = 42 - 22 = 20

AP: 8, 14, 20, 26, ...  (d = 6)

General term: a(n) = S(n) - S(n-1)
= [3n^2 + 5n] - [3(n-1)^2 + 5(n-1)]
= 3n^2 + 5n - 3n^2 + 6n - 3 - 5n + 5
= 6n + 2

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6. Properties of Arithmetic Progression

Property 1: Constant Addition/Subtraction/Multiplication

If a, b, c, d, ... are in AP, then:

• a+k, b+k, c+k, d+k, ... are also in AP (same d)
• a-k, b-k, c-k, d-k, ... are also in AP (same d)
• ka, kb, kc, kd, ... are also in AP (new common difference = k*d)
• a/k, b/k, c/k, d/k, ... are also in AP (new common difference = d/k)

Property 2: Three Numbers in AP

If a, b, c are in AP, then:

b - a = c - b
2b = a + c
b = (a + c) / 2

Here, b is called the Arithmetic Mean of a and c.

Property 3: Selecting Terms in AP

When choosing unknown terms in AP, use symmetric forms to simplify:

Number of terms	Choose as	Common difference
3 terms	a-d, a, a+d	d
4 terms	a-3d, a-d, a+d, a+3d	2d
5 terms	a-2d, a-d, a, a+d, a+2d	d
6 terms	a-5d, a-3d, a-d, a+d, a+3d, a+5d	2d

Property 4: Sum Properties

• The sum of terms equidistant from the beginning and end is constant:

a(1) + a(n) = a(2) + a(n-1) = a(3) + a(n-2) = ... = constant

• If S(n) of an AP is given, then:

a(n) = S(n) - S(n-1)

Property 5: Middle Term

• If an AP has odd number of terms, the middle term equals the average of all terms:

Middle term = S(n) / n

Property 6: AP from Condition

• If a(n) = pn + q (linear in n), the sequence is always an AP with:

Common difference d = p
First term a = p + q

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7. Arithmetic Mean (AM)

Single Arithmetic Mean

The Arithmetic Mean of two numbers a and b is:

AM = (a + b) / 2

Inserting n Arithmetic Means Between Two Numbers

To insert n arithmetic means between a and b:

Total terms in the new AP = n + 2  (including a and b)

Common difference d = (b - a) / (n + 1)

The n arithmetic means are:
A(1) = a + d
A(2) = a + 2d
A(3) = a + 3d
...
A(n) = a + nd

Example: Insert 4 arithmetic means between 3 and 23.

a = 3, b = 23, n = 4

d = (23 - 3) / (4 + 1) = 20 / 5 = 4

A(1) = 3 + 4 = 7
A(2) = 3 + 8 = 11
A(3) = 3 + 12 = 15
A(4) = 3 + 16 = 19

Complete AP: 3, 7, 11, 15, 19, 23

Sum of n Arithmetic Means

The sum of n arithmetic means inserted between a and b equals:

Sum of n AMs = n * (a + b) / 2

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8. Useful Results and Special Cases

Result 1: nth Term from the End

The nth term from the end of a finite AP:

a(n from end) = l - (n - 1) * d

Where l is the last term.

Result 2: Number of Terms

If the AP has first term a, common difference d, and last term l:

Number of terms n = [(l - a) / d] + 1

Result 3: Sum of AP with Equal Number of Terms

If two APs have the same number of terms:

Sum of combined AP = Sum of AP1 + Sum of AP2

Result 4: Condition for Three Numbers in AP

Three numbers a, b, c are in AP if and only if:

2b = a + c

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9. Real-Life Applications of AP

1. Simple Interest

Simple interest accumulation forms an AP:

Year 1: P + SI
Year 2: P + 2*SI
Year 3: P + 3*SI
...

Where P is principal and SI is simple interest per year.

2. Salary Increments

A person earning Rs. 20,000 with an annual increment of Rs. 1,500:

Year 1: 20,000
Year 2: 21,500
Year 3: 23,000
...
This is an AP with a = 20,000 and d = 1,500

3. Seating Arrangement

A theater where each row has 2 more seats than the previous row:

Row 1: 20 seats
Row 2: 22 seats
Row 3: 24 seats
...
AP with a = 20, d = 2

4. Distance Covered (Uniform Acceleration)

An object starting from rest under uniform acceleration covers distances in successive seconds:

1, 3, 5, 7, 9, ... (in some unit)
This is an AP with a = 1, d = 2

5. Stacking Problems

Logs stacked in layers where each layer has one fewer log:

Layer 1 (bottom): 15 logs
Layer 2: 14 logs
Layer 3: 13 logs
...
AP with a = 15, d = -1

6. Depreciation (Straight Line Method)

Asset value decreasing by a fixed amount each year:

Year 0: Rs. 100,000
Year 1: Rs. 90,000
Year 2: Rs. 80,000
...
AP with a = 100,000, d = -10,000

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10. Summary of All Formulas

Formula	Expression
Common difference	d = a(n) - a(n-1)
nth term (general term)	a(n) = a + (n-1)d
Sum of n terms (Form 1)	S(n) = n/2 * [2a + (n-1)d]
Sum of n terms (Form 2)	S(n) = n/2 * (a + l)
nth term from sum	a(n) = S(n) - S(n-1), n >= 2
Arithmetic Mean of a, b	AM = (a + b) / 2
n AMs between a and b	d = (b - a) / (n + 1)
Sum of n AMs between a, b	n * (a + b) / 2
Number of terms	n = [(l - a) / d] + 1
nth term from the end	l - (n - 1)d
Condition for AP	2b = a + c for three terms a, b, c
Sum of first n naturals	n(n + 1) / 2
Sum of first n even numbers	n(n + 1)
Sum of first n odd numbers	n^2

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