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Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers where each term increases or decreases by the same fixed amount, called the common difference.

General Term:

$$ a_n = a_1 + (n - 1)\cdot d $$ where:
an = nth term
a1 = first term
d = common difference

Generalized Formula:

This generalized formula is applicable in most arithmetic progression problems and is much more flexible.

$$ a_n = a_m + (n - m)\cdot d $$ where:
$a_n =$ succeeding term
$a_m =$ preceding term

Sum of the First n Terms:

$$ S_n = \frac{n}{2}(a_1 + a_n) $$

This formula works for any AP — whether the number of terms is even or odd — because it relies on a clever pairing strategy.


How the Pairing Works (Using 2, 4, 6, 8, 10):

Let's add the AP: 2, 4, 6, 8, 10.

We can pair the first and last terms:

The middle term (6) is left unpaired, but we can treat it as “paired with itself.”

So the total sum is:

$$ S = 12 + 12 + 6 = 30 $$

Or, using the formula:

$$ S_5 = \frac{5}{2}(2 + 10) = \frac{5}{2}(12) = 30 $$

The formula works because it essentially averages the first and last term, then multiplies by how many terms there are.

Why It Always Works:

In an AP, the first and last terms always sum to the same value as the second and second-to-last, third and third-to-last, and so on. Whether the number of terms is even or odd, this symmetry lets us group and multiply efficiently — which is exactly what the formula captures.

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Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference

The sixth term of an arithmetic progression is 17 and the thirteenth term is 38. Find the nineteenth term. Ans. 56

Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram

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Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram Arithmetic Progression | Algebra – Problem 1: Arithmetic Sequence CALTECH for any nth term and common difference – Diagram

Problem 2: Arithmetic Sequence CALTECH for the sum of the series

What is the sum of all odd integers between 7 and 777?

Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram

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Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram Arithmetic Progression | Algebra – Problem 2: Arithmetic Sequence CALTECH for the sum of the series – Diagram

Problem 3:

The 12th term, the 14th term, and the last term of an arithmetic sequence are 25, 31, and 37, respectively. Find the first term, common difference and the number of terms.

Arithmetic Progression | Algebra – Problem 3: – Diagram Arithmetic Progression | Algebra – Problem 3: – Diagram Arithmetic Progression | Algebra – Problem 3: – Diagram

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Arithmetic Progression | Algebra – Problem 3: – Diagram Arithmetic Progression | Algebra – Problem 3: – Diagram Arithmetic Progression | Algebra – Problem 3: – Diagram Arithmetic Progression | Algebra – Problem 3: – Diagram

Problem 4:

Find the number of terms and the sum of the terms of the arithmetic progression 32, 28, …4

Arithmetic Progression | Algebra – Problem 4: – Diagram Arithmetic Progression | Algebra – Problem 4: – Diagram Arithmetic Progression | Algebra – Problem 4: – Diagram

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Arithmetic Progression | Algebra – Problem 4: – Diagram Arithmetic Progression | Algebra – Problem 4: – Diagram Arithmetic Progression | Algebra – Problem 4: – Diagram Arithmetic Progression | Algebra – Problem 4: – Diagram

Problem 5:

In a pile of logs, each layer contains one more log than the layer above the top and the top contains just one log. If there are 105 logs in the pile, how many layers are there?

Arithmetic Progression | Algebra – Problem 5: – Diagram Arithmetic Progression | Algebra – Problem 5: – Diagram Arithmetic Progression | Algebra – Problem 5: – Diagram

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Arithmetic Progression | Algebra – Problem 5: – Diagram Arithmetic Progression | Algebra – Problem 5: – Diagram Arithmetic Progression | Algebra – Problem 5: – Diagram Arithmetic Progression | Algebra – Problem 5: – Diagram
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