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Revision Notes: Geometric Progression | Mathematics Class 10 ICSE PDF Download
Important Terms
- A sequence is an arrangement of numbers in a definite order according to some rule.
- The various numbers occurring in a sequence are called its terms. We denote the terms of a sequence by a1, a2, a3… etc. Here, the subscripts denote the positions of the terms. In general, the number at the nth place is called the nth term of the sequence and is denoted by an. The nth term is also called the general term of the sequence.
- A sequence having a finite number of terms is called a finite sequence.
- A sequence which do not have a last term and which extends indefinitely is known as an infinite sequence.
Geometric Progression
A sequence is said to be in geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. Constant Ratio is common ratio denoted by ‘r’.
General Term of a G.P.
The general term of a G.P. is given by
tn = arn –1
where ‘a’ is the first term and ‘r’ is the common ratio.
Properties of G.P.
- The ratio between the consecutive terms of a G.P. is always the same.
- rth term from the beginning x rth term from the end = constant = First term x Last term
- If a, b and c are in G.P,
- In a G.P. if the terms at equal distances are taken, these terms are also in G.P.
- If each term of a G.P. be multiplied or divided by the same non-zero number, the resulting series is also a G.P.
- The series obtained by taking the reciprocals of the terms of a G.P. is also a G.P.
- If each term of a G.P. is raised to the same non-zero number, the resulting series is also a G.P.
- If the corresponding terms of two different G.P.s are multiplied/divided together, the resulting series, so obtained, is also a G.P.
Sum of n terms of a G.P.
Sn = a + ar + ar² + … + arn-1
Case I: r = 1
Sn = a + a + a + … + to n terms = na
Case II: r < 1
Case III: r > 1
Sum of Infinite terms of a G.P.
Sum of infinite terms in G.P. , =
if |r| < 1Geometric Mean Between Numbers a and b
If a and b are two positive numbers then a, G, b are in G.P.
⇒ G2 = ab
⇒ G = √(ab)
Three or More Terms in G.P.
Number of terms is three: a/r, a, ar
Number of terms is four: a/(r3), a/r, a, ar3
Number of terms is five:
and so on.
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FAQs on Revision Notes: Geometric Progression - Mathematics Class 10 ICSE
| 1. What is a geometric progression (GP)? |  |
Ans. A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 6, 18, 54, the common ratio is 3 because each term is obtained by multiplying the previous term by 3.
| 2. How do you find the nth term of a geometric progression? | |
Ans. The nth term of a geometric progression can be found using the formula: \( a_n = a \cdot r^{(n-1)} \), where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. For example, if the first term is 5 and the common ratio is 2, the 4th term would be \( 5 \cdot 2^{(4-1)} = 5 \cdot 8 = 40 \).
| 3. What is the sum of the first n terms of a geometric progression? | |
Ans. The sum \( S_n \) of the first \( n \) terms of a geometric progression can be calculated using the formula: \( S_n = a \cdot \frac{(1 - r^n)}{(1 - r)} \) when \( r \neq 1 \), where \( a \) is the first term and \( r \) is the common ratio. For example, for a GP with a first term of 3 and a common ratio of 2, the sum of the first 4 terms would be \( 3 \cdot \frac{(1 - 2^4)}{(1 - 2)} = 3 \cdot \frac{(1 - 16)}{-1} = 3 \cdot 15 = 45 \).
| 4. Can a geometric progression have a negative common ratio? | |
Ans. Yes, a geometric progression can have a negative common ratio. In this case, the terms will alternate between positive and negative values. For example, if the first term is 4 and the common ratio is -2, the sequence will be 4, -8, 16, -32, and so on.
| 5. How can geometric progressions be applied in real life? | |
Ans. Geometric progressions are used in various real-life scenarios, such as calculating compound interest in finance, modeling population growth, or analyzing exponential decay in natural sciences. For instance, if an investment grows at a fixed percentage rate annually, the value of the investment over time can be represented as a geometric progression.
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