All forms of A.P.

All Important Formulas of Arithmetic Progression (A.P.)

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1. What is an Arithmetic Progression?

An Arithmetic Progression is a sequence in which the difference between consecutive terms is constant.

Example:
[2, 5, 8, 11, 14, …]

Here, the constant difference is 3.

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2. Basic Terms and Notations in A.P.

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3. Common Difference Formula

The common difference is the difference between any term and the previous term.

[ d = a_2 − a_1 ]

or

[ d = T_{n} − T_{n-1} ]

Explanation:
In an A.P., this value is the same for all consecutive terms.

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4. nᵗʰ Term (General Term) Formula

The formula for the nᵗʰ term of an Arithmetic Progression is:

[ T_{n} = a + (n − 1)d ]

When to use:

• To find any specific term (10ᵗʰ term, 20ᵗʰ term, etc.)
• To identify whether a sequence is an A.P.

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5. Sum of First n Terms of an A.P.

Formula 1 (when a and d are known)

[ S_{n} = \dfrac{n}{2} [2a + (n − 1)d] ]

Use this formula when:

• First term [a] and common difference [d] are given

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Formula 2 (when a and last term l are known)

[ S_{n} = \dfrac{n}{2} (a + l) ]

Use this formula when:

• First term [a] and last term [l] are given

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6. Last Term of an A.P.

The last term of an A.P. with [n] terms is:

[ l = a + (n − 1)d ]

Note:
This is actually the same as the [n^{th}] term formula.

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7. Arithmetic Mean (A.M.) Formula

Arithmetic Mean between two numbers

If two numbers are [a] and [b], then:

[ A.M. = \dfrac{a + b}{2} ]

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Inserting n Arithmetic Means between a and b

Common difference:

[ d = \dfrac{b − a}{n + 1} ]

The Arithmetic Means are:

[a + d, a + 2d, a + 3d, … , a + nd]

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8. nᵗʰ Term from the Sum of n Terms

If [Sₙ] is given, then the nᵗʰ term is:

[ T_{n} = S_{n} − S_{n-1} ]

Very Important:
If [S_{n}] is a quadratic expression, then the sequence is an A.P.

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9. Relationship between First Term, Last Term, and Number of Terms

[ n = \dfrac{l − a}{d} + 1 ]

Used when:

• First term, last term, and common difference are known
• Number of terms is to be found

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10. Middle Term of an A.P.

If an A.P. has an odd number of terms, then the middle term is:

[ T_{\dfrac{n+1}{2}} ]

Also,

[ \text{Middle term} = \dfrac{a + l}{2} ]

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11. Important Special Results

• If [a = 0], the A.P. starts from zero
• If [d = 0], all terms are equal
• If [d > 0], the A.P. is increasing
• If [d < 0], the A.P. is decreasing

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12. Summary Table (Very Useful for Exams)

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Exam Tip for Students

Before solving any A.P. problem, always:

1. Identify [a], [d], and [n]
2. Decide which formula fits the given data
3. Write the formula before substituting values

This habit alone can double accuracy in exams.