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Questions and Answers
In a Fibonacci sequence, what is the value of $a_5$ given that $a_1 = 1$, $a_2 = 1$?
In a Fibonacci sequence, what is the value of $a_5$ given that $a_1 = 1$, $a_2 = 1$?
- 2
- 3
- 8 (correct)
- 5
Which of the following is a correct representation of a Geometric Progression (G.P.)?
Which of the following is a correct representation of a Geometric Progression (G.P.)?
- 1, 3, 6, 10, 15
- 2, 5, 8, 11, 14
- 1, 2, 4, 8, 16
- 9, 27, 81, 243 (correct)
If the nth term of a sequence is defined by the formula $a_n = a_{n-1} - 1$ for $n > 2$, and if $a_2 = 2$, what is the value of $a_4$?
If the nth term of a sequence is defined by the formula $a_n = a_{n-1} - 1$ for $n > 2$, and if $a_2 = 2$, what is the value of $a_4$?
- 3
- 1
- 2
- 0 (correct)
In a sequence where $a_1 = -1$ and the recursive formula is given by $a_n = rac{a_{n-1}}{n}$ for $n ≥ 2$, what is the value of $a_3$?
In a sequence where $a_1 = -1$ and the recursive formula is given by $a_n = rac{a_{n-1}}{n}$ for $n ≥ 2$, what is the value of $a_3$?
Which statement correctly describes the relationship between the Fibonacci sequence and the growth of natural numbers?
Which statement correctly describes the relationship between the Fibonacci sequence and the growth of natural numbers?
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Study Notes
Introduction
- Natural numbers are a product of human spirit – Dedekind
- Sequences can be used to represent patterns and relationships in data
- Series are used to represent the sum of a sequence's terms
Types of Sequences and Series
- Arithmetic Progression (A.P.)
- Defined by a common difference between consecutive terms
- Example: 2, 4, 6, 8...where the common difference is 2
- Geometric Progression (G.P.)
- Common ratio between consecutive terms
- Example: 2, 4, 8, 16… where the common ratio is 2
Fibonacci Sequence
- Defined by the sum of the previous two terms
- First two terms are 1, 1
- Example: 1, 1, 2, 3, 5, 8, 13…
- Formula: an = an–1 + an–2, n > 2
- Golden ratio (phi): approximately 1.618
- The ratio of two consecutive Fibonacci numbers approaches phi as the numbers increase
Geometric Progression (G.P.)
- Example: 2, 4, 8, 16…
- Each term is found by multiplying the previous term by a common ratio
- (In this case, the common ratio is 2)
- Examples:
- 2, 4, 8, 16…
- 1/9, 1/27, 1/81, 1/243
- .01, .0001, .000001…
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