01 · Explore
Understanding Sequences
In mathematics, a sequence is an ordered collection of objects or numbers where each member has a specific position.
A sequence is formally defined as a function whose domain is the set of natural numbers (1, 2, 3, ...) or a finite subset of it. The individual numbers in a sequence are called its terms, denoted by a₁, a₂, a₃, ..., aₙ. The subscript indicates the position of the term, and aₙ is referred to as the general term.
Sequences can be finite or infinite. A finite sequence has a fixed number of terms, such as the number of ancestors a person has over ten generations (2, 4, 8, ..., 1024). An infinite sequence continues indefinitely, like the successive quotients in the division of 10 by 3 (3, 3.3, 3.33, ...).
While many sequences follow a specific algebraic formula, others are defined by recurrence relations where a term depends on previous terms. A famous example is the Fibonacci sequence, where a₁ = a₂ = 1 and aₙ = aₙ₋₂ + aₙ₋₁ for n > 2.
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What is the difference between a finite and an infinite sequence?
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Define the Fibonacci sequence using a recurrence relation.
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02 · Explore
Series and Sigma Notation
A series is the expression formed by adding the terms of a sequence.
If a₁, a₂, a₃, ..., aₙ is a sequence, then the expression a₁ + a₂ + a₃ + ... + aₙ is the associated series. It is important to distinguish between the 'series' (the indicated sum) and the 'sum of the series' (the numerical result of the addition).
To represent series compactly, mathematicians use the Greek letter Σ (sigma). The notation Σ aₖ from k=1 to n represents the sum of terms from the first to the nth position.
Finding Terms from a General Formula
aₙ = (n - 1)(2 - n)(3 + n)
To find the 20th term (a₂₀), substitute n = 20 into the formula: a₂₀ = (20 - 1)(2 - 20)(3 + 20) = 19 × (-18) × 23 = -7866.
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If a sequence is 2, 4, 6, 8, what is the corresponding finite series?
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03 · Explore
Geometric Progression (G.P.) Basics
A Geometric Progression is a sequence where each term (except the first) bears a constant ratio to the preceding term.
In a G.P., the ratio of any term to its predecessor is constant and is called the common ratio (r). The first term is usually denoted by 'a'. For a sequence to be a G.P., every term must be non-zero.
The general form of a G.P. is a, ar, ar², ar³, ..., arⁿ⁻¹. For example, in the sequence 2, 4, 8, 16, the first term a = 2 and the common ratio r = 4/2 = 2.
| Sequence | First Term (a) | Common Ratio (r) | Type |
|---|---|---|---|
| 2, 4, 8, 16... | 2 | 2 | G.P. |
| 1/9, -1/27, 1/81... | 1/9 | -1/3 | G.P. |
| 0.01, 0.0001, 0.000001... | 0.01 | 0.01 | G.P. |
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How do you calculate the common ratio 'r' of a G.P.?
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04 · Explore
The General Term of a G.P.
The nth term of a geometric progression can be found using a specific algebraic formula derived from the first term and common ratio.
By observing the pattern of terms: a₁ = a, a₂ = ar¹, a₃ = ar², we see that the exponent of 'r' is always one less than the position of the term 'n'.
The formula for the nth term (general term) is aₙ = arⁿ⁻¹. This formula allows us to find any specific term in a G.P. without listing all preceding terms.
Finding a Specific Term
aₙ = 5(5)ⁿ⁻¹
In the G.P. 5, 25, 125, ..., a = 5 and r = 5. To find the 10th term: a₁₀ = 5(5)¹⁰⁻¹ = 5(5)⁹ = 5¹⁰.
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Find the 9th term of a G.P. where a = 2 and r = 4.
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05 · Explore
Sum to n Terms of a G.P.
The sum of the first n terms of a Geometric Progression (Sₙ) depends on whether the common ratio is equal to or different from 1.
If the common ratio r = 1, the G.P. is a, a, a, ... and the sum Sₙ = na.
If r ≠ 1, the sum is given by the formula Sₙ = a(1 - rⁿ) / (1 - r) or Sₙ = a(rⁿ - 1) / (r - 1). These two forms are mathematically equivalent; the first is often used when |r| < 1 and the second when |r| > 1 to keep the denominator positive.
Calculating the Sum of a Series
S₅ = 3[1 - (2/3)⁵] = 3[1 - 32/243] = 3 × 211/243 = 211/81
For the series 1 + 2/3 + 4/9 + ..., a = 1 and r = 2/3. For n = 5, the sum is calculated using Sₙ = a(1 - rⁿ) / (1 - r). Substituting the values: S₅ = 1(1 - (2/3)⁵) / (1 - 2/3) = (1 - 32/243) / (1/3) = 3(211/243) = 211/81.
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What is the sum formula for a G.P. when r = 1?
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06 · Explore
Geometric Mean (G.M.)
The Geometric Mean of two positive numbers is the square root of their product.
For two positive numbers 'a' and 'b', the G.M. is √(ab). If we insert this value between a and b, the resulting sequence a, √(ab), b forms a G.P.
We can insert multiple numbers G₁, G₂, ..., Gₙ between a and b such that the entire sequence a, G₁, G₂, ..., Gₙ, b is a G.P. In this case, b is the (n+2)th term of the progression.
Inserting n Geometric Means
- 1
Identify Terms
Let 'a' be the first term and 'b' be the (n+2)th term.
- 2
Find Ratio
Calculate common ratio r = (b/a)^(1/(n+1)).
- 3
Generate Means
Calculate G₁ = ar, G₂ = ar², ..., Gₙ = arⁿ.
The process of inserting n geometric means between two numbers to create a G.P.
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Find the G.M. of 2 and 8.
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07 · Explore
Relationship Between A.M. and G.M.
There is a fundamental inequality relating the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of two positive numbers.
For two positive real numbers a and b, the Arithmetic Mean A = (a + b)/2 and the Geometric Mean G = √(ab).
The relationship states that A ≥ G. This can be proved by observing that A - G = (a + b)/2 - √(ab) = (√a - √b)² / 2. Since the square of any real number is non-negative, A - G ≥ 0, implying A ≥ G.
Finding Numbers from A.M. and G.M.
a + b = 20; ab = 64
If A.M. = 10 and G.M. = 8, then (a+b)/2 = 10 and √(ab) = 8. Using (a-b)² = (a+b)² - 4ab: (a-b)² = 400 - 256 = 144. So a-b = 12. Solving a+b=20 and a-b=12 gives a=16, b=4.
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Is it possible for the G.M. of two distinct positive numbers to be greater than their A.M.?
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08 · Explore
Practical Applications of Progressions
Sequences and series are used to model real-world growth, depreciation, and financial calculations.
Geometric progressions are particularly useful for modeling population growth (like bacteria doubling every hour) or financial interest that is compounded annually.
Depreciation of assets, where a value decreases by a fixed percentage every year, also follows a geometric pattern. For example, if a machine worth Rs 15625 depreciates by 20% annually, its value at the end of each year forms a G.P. with r = 0.80.
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If a bank pays 10% interest compounded annually on Rs 500, what is the common ratio 'r' for the amount at the end of each year?
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