Arithmetic and Geometric Progressions

Arithmetic and Geometric Progressions

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In this chapter, we will first get familiarised with progressions and how they are involved in our daily lives. Next, we proceed to the basic terminologies used. General examples are given for better understanding. The unit next to this focuses on the nth term of the sequence - the process of finding it and the general formula of the same are presented. The focus then shifts to finding the sum of these terms in a progression. We see how the formula is derived, and then examples are presented to enhance students’ understanding. The section following this is related to geometric progression and its examples. Sums of finite and infinite GP are also explained.

In our study of Mathematics, we will encounter various kinds of progressions, which are sequences of numbers with a definite relationship between successive numbers. Let us consider a few examples of progressions.

Consider the following sequence of numbers:

\[ - 3,\, - 1,\,\,1,\,\,3,\,\,5,\,...\]

What do you notice about this sequence? The most important characteristic of this sequence is that the difference between successive terms is constant, and equal to 2. This characteristic enables us to say that the next term in this progression will be 7. This sequence is an example of an arithmetic progression. Here is another example of an arithmetic progression:

\[17,\;14,\;11,\;8,\;5,...\]

The difference between successive terms in this progression is –3, and the next term in the sequence will be 2.

Now, consider the following sequence of numbers:

\[2,\,4,\,8,\,16,\,32,...\]

In this sequence, each term is double the previous term. In other words, the ratio of successive terms is constant, and equal to 2. This sequence is an example of a geometric progression. In any geometric progression, the ratio of successive terms is constant. Here is another example:

\[1,\,\frac{1}{2},\,\frac{1}{4},\,\frac{1}{8},\,\frac{1}{{16}},...\]

Now, consider the following sequence:

\[\frac{1}{3},\,\frac{1}{5},\,\frac{1}{7},\,\frac{1}{9},\,\frac{1}{{11}},...\]

The important characteristic of this sequence is that the reciprocal of its terms are in arithmetic progression:

\[3,\,\,5,\,\,7,\,\,9,\,\,11,...\]

Such sequences, in which the reciprocal terms are in arithmetic progressions, are called harmonic progressions. Here is another example of a harmonic progression:

\[\frac{2}{7},\,\,\frac{2}{{11}},\,\,\frac{2}{{15}},\,\,\frac{2}{{19}},\,\,\frac{2}{{23}},...\]

Verify that its reciprocal terms are in arithmetic progression.

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