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 Tn = a .rn-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, ar2, …. In which the nth term Tn = a. rn-1 is called the general term of the geometric progression.
nth term of Geometric Progression:
The nth term of geometric progression a, ar, ar2, … is Tn = a. rn-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: Tn = a. rn-1
Meaning of Geometric Series – Sequence:
If a geometric progression is expressed as a, ar, ar2, ar3, … in terms of a and r, then a + ar + ar2 + ar3 + … + arn-1 is called geometric series. It is denoted by Sn i.e.,
= a + ar + ar2 + … + a. rn-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.
Sn = T1 + T2 + T3 +… + Tn Where, TR = q.rn-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, ar3
- For n = 5, terms are : \(\frac{a}{r^{2}}, \frac{a}{r}\), a, ar, ar2
Symbols:
- a = First term of the progression
- r = Common ratio of the progression
- Tn = nth term of the progression
- Sn = 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. Tn+1 = (n + 1 )th term
Tn = nth term ………(1)
2. Sequence formula :
Tn = a. rn-1
Where, a = First term
r = Common ratio…………(2)
n = Serial of the term
3. Series formula :
Sn = n . a
Where, r = 1 …….(3)
- Sn = \(\frac{a\left[1-r^{n}\right]}{1-r}\) Where, |r| < 1 ……..(4)
- Sn = \(\frac{a\left[r^{n}-1\right]}{r-1}\) Where, |r| > 1 ……..(5)
- Sn = \(\frac{a-r \mathrm{~T}_{n}}{1-r}\) Where, |r| < 1 ………(6)
- Sn = \(\frac{r T_{n}-a}{r-1}\) Where, |r| > 1 ……….(7)
4. Other formulae:
Tn+1 = Sn+1 – Sn, n = 1, 2, 3………….. (8)
Where, Sn+1 = Sum of first (n + 1) terms,
Sn = 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, ar3;
Where, common ratio = r2 - 5 terms: \(\frac{a}{r^{2}}, \frac{a}{r}\), a. ar, ar2;
Where common ratio = r