PUMPA - SMART LEARNING

எங்கள் ஆசிரியர்களுடன் 1-ஆன்-1 ஆலோசனை நேரத்தைப் பெறுங்கள். டாப்பர் ஆவதற்கு நாங்கள் பயிற்சி அளிப்போம்

Book Free DemoLike how we have dealt with arithmetic progression, we will deal with geometric progression in this section.

Situation 1

In this situation, the midpoints of \(\triangle ABC\) is joined to form \(\triangle DEF\).

Similarly, the midpoint of \(\triangle CEF\) is joined to form \(\triangle MNO\).

The midpoints of \(\triangle CON\) is joined to form a different triangle and so on.

The area of \(\triangle ABC\), \(\triangle DEF\), \(\triangle MNO\) and so on is given by \(\triangle ABC\), \(\frac{1}{4}\triangle ABC\), \(\frac{1}{4} \times \frac{1}{4} \triangle ABC\) and so on.

That is, \(\triangle ABC\), \(\frac{1}{4} \triangle ABC\), \(\frac{1}{16} \triangle ABC\), and so on.

In other words, the areas of \(\triangle ABC\), \(\triangle DEF\), \(\triangle MNO\) and so on are \(\frac{1}{4}\) apart.

Therefore, the areas of the triangles formed are in a geometric progression with \(\frac{1}{4}\) as the common ratio.

Situation 2

A particular dog breed gives birth to exactly two puppies at a time.

The number of dogs in each level is indicated as \(1\), \(2\), \(4\),\(…\)

Therefore, the number of dogs is a geometric progression with \(2\) as the common ratio.

Geometric progression

A Geometric Progression is a sequence in which each term is obtained by multiplying a fixed non-zero number by the preceding term except the first term. The fixed number is called the common ratio. The common ratio is usually denoted by \(r\).

**The general form of geometric progression**:

A geometric progression is given in the form of \(a\), \(ar\), \(ar^2\),\(...ar^{n-1}\).

Here, \(a\) is the first term, and \(r\) is the common ratio.

The first term \(a\), when multiplied by the \(r\) subsequently, forms \(a\), \(ar\), \(ar^2\),\(...ar^{n-1}\), which is the geometric progression.