Arithmetic Progression on GMAT

50-Word Summary

An arithmetic progression is a sequence where each term changes by a fixed common difference. For example, 2, 5, 8, 11 has difference 3. The nth-term is a + (n-1)d, while the sum of n terms is given by Sₙ = n/2 [2a + (n-1)d].

Overview

Arithmetic progressions are practical and elegant, and they often feature on the GMAT. An arithmetic progression is a sequence where each term differs from the previous one by a constant amount, forming the basis of many structured problems. Early mastery during GMAT prep makes them powerful, as their simplicity leads to strong relationships that save time. The ability to identify the nth term or quickly calculate a sum without step-by-step addition highlights the importance of recognizing patterns. What the test truly evaluates is not formula memorization but the clarity and discipline to see order in repeated structures.

 

Understanding Arithmetic Progressions on the GMAT

An arithmetic progression (AP) is a sequence of numbers in which each term differs from the previous one by a constant amount, known as the common difference. Examples include 1, 2, 3, 4 with a common difference of +1, or 9, 7, 5, 3 with a common difference of −2.

Useful Relationships

1. The nth term of a series

The nth term of an arithmetic progression is given by:
tn = a + (n − 1)d
where a = first term, d = common difference, n = term position.

2. The sum of the first n terms of a series

The sum of the first n terms (Sn) can be found using:
Sn = (n/2) × [2a + (n − 1)d]
or equivalently,

Sn = (n/2)(a + l), where l = last term.

Useful Outcomes

• Sum of the first n positive integers = n(n + 1)/2
• Sum of the first n positive even integers = n(n + 1)
• Sum of the first n positive odd integers = n²

Examples

1.Find the 20th term when a = 5 and d = 5.
tn = a + (n − 1)d
So, t20 = 5 + (20 − 1) × 5 = 5 + 95 = 100

 

2.Find the sum of the first 50 terms when a = 3 and d = 3.

Sn = (n/2) × [2a + (n − 1)d]
So, S50 = (50/2)[2 × 3 + (50 − 1) × 3] = 25(6 + 147) = 25 × 153 = 3825

 

3.Find the sum of the first 25 positive integers.
Sum of the first n positive integers = n(n + 1)/2

So, S25 = 25 × 26/2 = 325

 

4.Find the sum of the first 20 positive even integers.
Sum of the first n positive even integers = n(n + 1)

So, S20 = 20 × 21 = 420

 

5.Find the sum of the first 30 positive odd integers.

Sum of the first n positive odd integers = n²
So, S30 = 30² = 900

The Larger Insight

Arithmetic progressions highlight the elegance of structure in numbers. They show how repetition creates patterns, and how recognizing those patterns brings efficiency. The GMAT uses them not to check memorization but to measure clarity of thought. Regular practice through GMAT mocks strengthens reasoning, builds confidence, and ensures that even with large numbers, solutions remain clear and systematic.

Parting Thought

Arithmetic progressions remind us that life often advances in steady steps, with each stage building naturally upon the last. Success in GMAT preparation also comes from recognizing and trusting such progress—small, consistent efforts that accumulate into mastery. The MBA applications process reflects a similar rhythm, where essays, recommendations, and interviews align to form a clear story of growth. In life, too, patterns emerge through persistence and discipline, turning repetition into strength and structure into opportunity. The real lesson is that patience with gradual progress unlocks profound results, both in exams and in the larger journeys that define our paths.