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Arithmetic Progressions — nth Term, Sum, Examples

TL;DR

An arithmetic progression (AP) is a sequence where each term increases by a fixed amount called the common difference d, like 2, 5, 8, 11,… This article covers the definition, the nth-term formula an = a + (n−1)d, both sum formulas, a quick derivation, six worked examples, and the mistakes students make most.

What is an Arithmetic Progression?

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. That constant is the common difference, written d. Each new term is the one before it plus d.

A general AP looks like this:

a, a + d, a + 2d, a + 3d, …

where a is the first term. Note that d can be negative (a decreasing AP) or zero (a constant sequence). It just has to be the same every step.

The Arithmetic Progression Formulas

• nth term (general term):

  an = a + (n−1)d

• Sum of the first n terms (when you know a and d):

  Sn = n/2 [2a + (n−1)d]

• Sum of the first n terms (when you know the first and last terms):

  Sn = n/2(a + l)

The two sum formulas are based on the same idea, just formulated differently.

Where the nth-term formula comes from

It is based on repeated addition: the first term stays the same, the second term is the first term plus d, the nth term accumulates d (n−1) times.

Where the sum formula comes from (Gauss's trick)

Write the sum in reverse; pair terms, and you can derive the sum formula.

Examples of Arithmetic Progressions

Example 1

Find the common difference of 5, 8, 11, 14,…

Subtract consecutive terms: 8−5=3 and 11−8=3.

Final answer: d = 3.

Example 2

Find the 10th term of 7, 12, 17, 22,…

Correctly applying the formula:

a10 = a + (n−1)d = 7 + (10−1)(5) = 52.

Final answer: 52.

Example 3

Which term of 3, 8, 13, 18,… equals 98?

Solve for n:

98 = 3 + (n−1)(5) \n95 = (n−1)(5) \nn = 20.

Final answer: the 20th term.

Example 4

Find the sum of the first 20 terms of 4, 9, 14, 19,…

Using the sum formula:

S20 = 20/2 [2(4) + (20−1)(5)] = 1030.

Final answer: 1030.

Example 5

Add all the integers from 1 to 100.

This sequence is an AP:

S100 = 100/2(1 + 100) = 5050.

Final answer: 5050.

Example 6

A starting salary is 30,000, rising by 2,000 each year. What is the total earned over 10 years?

Using the sum formula:

S10 = 10/2 [2(30000) + (10−1)(2000)] = 390000.

Final answer: 390,000 earned over the decade.

Where Arithmetic Progressions Show Up

The constant-step pattern is seen in:

• Salaries and savings
• Loan repayment and depreciation
• Seating, stacking, and scheduling

The pairing trick that powers the sum formula is usually credited to Carl Friedrich Gauss.

Common Mistakes With Arithmetic Progressions

Mistake 1: Using n instead of (n−1) in the nth-term formula.

Mistake 2: Getting the common difference backwards.

Mistake 3: Picking the wrong sum formula.

Conclusion

An arithmetic progression adds a constant common difference at every step. The nth term and the sum formulas are crucial to solving AP problems efficiently.