Arithmetic and Multiplicative Progressions (AP, GP).

One could expect one or two questions based on the topic sequences and series. The typical series tested included arithmetic progression and geometric or multiplicative progression. 4GMAT's GMAT Math Lesson Book in this chapter covers the following concepts in the topic sequences and progressions.

1. Introduction to Arithmetic Progression
2. Explanation with formulae to find the nth term of an arithmetic progression and the sum of n terms of an arithmetic progression.
3. Illustrative and solved examples to find the value of the common difference, first term and the number of terms, given the sum of n terms or the nth term and first term of an AP.
4. Property changes when an AP is transformed by addition, multiplication of constant terms.
5. Introduction to Multiplicative Progression (Geometric Progression)
6. Formulae to find the nth term of a multiplicative progression and the sum up to n terms of a multiplicative progression.
7. Introduction to the concept of infinitely decreasing multiplicative progression and the formulae to find the sum of such a sequence.
8. Geometric Mean or the mean of the Multiplicative progression.
9. Transformation witnessed when a Multiplicative Progression is multiplied by a constant.
10. Introduction to Harmonic progression. Formulae to find the nth term of a Harmonic Progression.
11. Relation between Arithmetic mean, Multiplicative mean and Harmonic mean.
12. 2 illustrative examples to explain concepts; 25 solved examples (with shortcuts wherever applicable) to acquaint you with as many different questions as possible; around 20 exercise problems with answer key and explanatory answers to provide you with practice and an objective type speed test with 40+ questions. Explanatory answers and answer key are provided for the speed test.

Here is an example of a typical solved example in this chapter.
Sample Question
Question
There are 4 terms in an A.P. such that the sum of the two means is 21 and the product of the extremes is 54. What are the terms of the A.P?

Explanatory Answer
Let the four terms be a - 3d, a - d, a + d and a + 3d.

The sum of the two means = a - d + a + d = 2a = 21 or a = 10.5

The product of the two extremes = (a - 3d)(a + 3d) = a2 - 9d2 = 54 => 10.52 - 9d2 = 54

=> 9d2 = 110.25 - 54 = 56.25
=> 3d = 7.5 => d = 2.5.

The four terms are a - 3d = 10.5 - 7.5 = 3, a - d = 10.5 - 2.5 = 8; a + d = 10.5 + 2.5 = 13 and a + 3d = 10.5 + 7.5 = 18. ie. 3, 8, 13, 18

Note
In the above expression, the term 'a' is not the first term as is generally assumed and the common difference is not 'd'. There is actually no term as 'a' as part of this progression. The terms are a - 3d, a - d ... and the common difference is 2d.