GMAT Maths | Sequences Series Q16

The GMAT Sample Math question given below is from the topic Arithmetic Progressions. Concept: Sum of an arithmetic progression. Problems on Arithmetic Progressions are quite easy if one understands that the basic concept behind AP is just an extrapolation of simple multiplication tables.

Question 16: What is the sum of all 3 digit numbers that leave a remainder of '2' when divided by 3?

1. 897
2. 164,850
3. 164,749
4. 149,700
5. 156,720

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Video Explanation

Play Video: Sum of Arithmetic Progressions Practice Question

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Explanatory Answer

Step 1 of solving this GMAT Progressions Question:
Identify the series

The smallest 3 digit number that will leave a remainder of 2 when divided by 3 is 101.
The next couple of numbers that will leave a remainder of 2 when divided by 3 are 104 and 107.
The largest 3 digit number that will leave a remainder of 2 when divided by 3 is 998.
It is evident that the numbers in the sequence will be a 3 digit positive integer of the form (3n + 2).

So, the given numbers are in an Arithmetic Progression with 101 as the first term, 998 as the last term, and 3 as the common difference of the sequence.

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Step 2 of solving this GMAT Progressions Question:
Compute the sum

Sum of an Arithmetic Progression (AP) = \\left[\frac{\text{first term + last term}}{2}\right]n), where 'n' is the number of terms in the sequence.

We know the first term: 101
We know the last term: 998.
The only unknown is the number of terms, n.

In an A.P., the nth term a_n = a₁ + (n - 1) * d
In this question, 998 = 101 + (n - 1) * 3
Or 897 = (n - 1) * 3
(n - 1) = 299 or n = 300.

Sum of the AP is \\left[\frac{101 + 998}{2}\right] * 300) = 164,850

Choice B is the correct answer.

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Key Concepts in Arithmetic Progressions

What is an Arithmetic Progression?

An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant.

An arithmetic progression, also known as an arithmetic sequence, is characterized by a consistent increase or decrease between consecutive terms. This constant difference is often called the "common difference" and is typically denoted by the letter 'd'.

The general form of an arithmetic progression can be written as:
a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d

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What are the main terms denoted in Arithmetic Progression?

First Term (a): This is the initial term from which the progression begins.
Common Difference (d): It is the fixed difference between consecutive terms in the progression.
Nth Term (an): Refers to any term's position in the sequence, calculated based on the first term and the common difference.

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What is the general formula for calculating the nth term of an Arithmetic Progression (AP)?

The nth term of an AP can be calculated using the formula: an = a + (n - 1) × d

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Examples of terms in an Arithmetic Progression (Arithmetic Sequence)

Simple increasing sequence: 2, 5, 8, 11, 14, 17, ...
Here, the first term (a) is 2, and the common difference (d) is 3.

Decreasing sequence: 20, 15, 10, 5, 0, -5, ...
In this case, a = 20 and d = -5 (negative common difference).

Sequence with fractional common difference: 1, 1.5, 2, 2.5, 3, 3.5, ...
Here, a = 1 and d = 0.5.

Sequence starting with a negative number: -7, -3, 1, 5, 9, 13, ...
In this example, a = -7 and d = 4.

Constant sequence (special case where d = 0): 4, 4, 4, 4, 4, ...
Here, a = 4 and d = 0.

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What are the types of Arithmetic Progressions?

Based on the number of terms included, APs are categorized into two main types: Finite AP and Infinite AP.

1. Finite AP

A Finite Arithmetic Progression is characterized by a specific number of terms. It begins with a first term and increases by a common difference with each subsequent term, but crucially, it concludes with a definitive last term. This boundary makes the sequence manageable and calculable when considering sums and limits.

Example: Consider the AP sequence 2, 4, 6, 8, 10. This sequence starts at 2 and increases by 2 each step, ending at 10.

2. Infinite AP

In contrast, an Infinite Arithmetic Progression does not have a terminal point. This type of AP keeps extending indefinitely, increasing by a fixed amount with each term but never concluding. Infinite APs are particularly useful in theoretical mathematics and offer intriguing insights when analyzing patterns.

Example: An AP starting at 1 and increasing by 3 might look like 1, 4, 7, 10, 13, ..., continuing endlessly.

Infinite Arithmetic Progressions questions are not typically tested in the Quantitative Reasoning Section of the GMAT.