This GSEB Class 11 Commerce Statistics Notes Chapter 9 Geometric Progression covers all the important topics and concepts as mentioned in the chapter.

## Geometric Progression Class 11 GSEB Notes

**Meaning of Geometric Progression:**

- Let us consider a sequence 2, 4, 8, 16,…. Ratio of consecutive terms of this sequence (\(\frac{4}{2}\) = 2, \(\frac{8}{4}\) = 2, …) is constant 2.
- Let us consider a sequence 3125, 625, 125, 25, …

Ratio of consecutive terms of this sequence \(\left(\frac{625}{3125}=\frac{1}{5}, \frac{125}{625}=\frac{1}{5}, \ldots\right)\) is constant \(\frac{1}{5}\) - Let us consider a sequence 3, -12, 48, – 192, …

Ratio of consecutive terms of this sequence (\(\frac{-12}{3}\) = -4, \(\frac{48}{-12}\) = – 4, … ) is constant -4. - For each of the above sequences, the ratio of (n + 1)th term and nth term (n ≥ 1) is non-zero constant. Such sequences are known as Geometric progression.

**Geometric Progression:**

If a and r non-zero real numbers, the sequence whose nth term is T_{n} = a .r^{n-1} for an integer n ≥ 1, is called geometric progression where, a = first term and r = common ratio.

**General Form of Geometric Progression:**

The general form of Geometric progress whose first term = a and the common ratio = r is a, ar, ar^{2}, …. In which the nth term T_{n} = a. r^{n-1} is called the general term of the geometric progression.

**nth term of Geometric Progression:**

The nth term of geometric progression a, ar, ar^{2}, … is T_{n} = a. r^{n-1} which is the general term of geometric progression.

**Common Ratio:**

In a geometric progression, for n ≥ 3= 1 if \(\frac{\mathrm{T}_{n+1}}{\mathrm{~T}_{n}}\) = r, then r is called the common raio of geometric progression. The value r can be any positive or negative integer or fraction except 0 and 1.

**Sequence Formula:**

The formula to find the nth term of geometric progression is called sequence formula. It is as follows: T_{n} = a. r^{n-1}

**Meaning of Geometric Series – Sequence:**

If a geometric progression is expressed as a, ar, ar^{2}, ar^{3}, … in terms of a and r, then a + ar + ar^{2} + ar^{3} + … + ar^{n-1} is called geometric series. It is denoted by S_{n} i.e.,

= a + ar + ar^{2} + … + a. r^{n-1} = Sum of first n terms of geometric progression

**Series Formula:**

A formula to find the sum of first n terms of geometric progression is called series formula.

S_{n} = T_{1} + T_{2} + T_{3 }+… + T_{n} Where, T_{R} = q.r^{n-1}, n = 1, 2, 3,…

**Consecutive Terms of Geometric Progression:**

Sometime, the sum and the product of some consecutive terms of a GP are given and these consecutive terms are to be found. In this situations, forms of the terms of GP are to be assumed in such a way that the calculation of these terms become simple. For some values of n, the assumptions are as follows :

- For n = 3, terms are : \(\frac{a}{r}\), a, ar
- For n = 4, terms are : \(\frac{a}{r^{3}}, \frac{a}{r}\), ar, ar
^{3} - For n = 5, terms are : \(\frac{a}{r^{2}}, \frac{a}{r}\), a, ar, ar
^{2}

**Symbols:**

- a = First term of the progression
- r = Common ratio of the progression
- T
_{n}= nth term of the progression - S
_{n}= Sum of first n terms of progression - n = Number of terms In the progression

**Formulae:**

1. Common ratio:

r = \(\frac{\mathrm{T}_{n+1}}{\mathrm{~T}_{n}}\)

Where. T_{n+1} = (n + 1 )th term

T_{n} = nth term ………(1)

2. Sequence formula :

T_{n} = a. r^{n-1}

Where, a = First term

r = Common ratio…………(2)

n = Serial of the term

3. Series formula :

S_{n} = n . a

Where, r = 1 …….(3)

- S
_{n}= \(\frac{a\left[1-r^{n}\right]}{1-r}\) Where, |r| < 1 ……..(4) - S
_{n}= \(\frac{a\left[r^{n}-1\right]}{r-1}\) Where, |r| > 1 ……..(5) - S
_{n}= \(\frac{a-r \mathrm{~T}_{n}}{1-r}\) Where, |r| < 1 ………(6) - S
_{n}= \(\frac{r T_{n}-a}{r-1}\) Where, |r| > 1 ……….(7)

4. Other formulae:

T_{n+1} = S_{n+1} – S_{n}, n = 1, 2, 3………….. (8)

Where, S_{n+1} = Sum of first (n + 1) terms,

S_{n} = Sum of first n terms

5. Assumption for the form of terms of G.P.:

- 3 terms: \(\frac{a}{r}\), a, ar;

Where, common ratio = r - 4 terms: \(\frac{a}{r^{3}}, \frac{a}{r}\), ar, ar
^{3};

Where, common ratio = r2 - 5 terms: \(\frac{a}{r^{2}}, \frac{a}{r}\), a. ar, ar
^{2};

Where common ratio = r