This MCQ module is based on: 8.1 Introduction
TOPIC 23 OF 45
8.1 Introduction
🎓 Class 11
Mathematics
CBSE
Theory
Ch 8 — Sequences and Series
⏱ ~15 min
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This mathematics assessment will be based on: 8.1 Introduction
Targeting Class 11 level in Sequences Series, with Advanced difficulty.
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8.1 Introduction
In mathematics, the word sequence is used in much the same way as it is in ordinary English. We say "ages of a family in increasing order form a sequence" or "the names of the months of the year form a sequence". A sequence? is simply an ordered list; in school mathematics the entries are usually numbers.
Sequences appear everywhere: depreciated values of an asset, the population of a town measured year by year, the resonance frequencies of a guitar string, the family tree of ancestors. Two structural relations among the terms — arithmetic (constant difference) and geometric (constant ratio) — are studied so much that they have their own names: arithmetic and geometric progressions. This chapter focuses on geometric progressions and the corresponding sums (called series?), with quick recap of the arithmetic case.
Leonardo of Pisa (Fibonacci)
c. 1170 – 1250 CE
Italian mathematician whose Liber Abaci (1202) introduced Hindu–Arabic numerals to Europe. He posed the now-famous "rabbit problem" whose solution is the recurrence \(a_n=a_{n-1}+a_{n-2}\) with \(a_1=a_2=1\) — producing the sequence 1, 1, 2, 3, 5, 8, 13, 21, … . The ratios of successive Fibonacci numbers approach the golden ratio \(\varphi=\dfrac{1+\sqrt 5}{2}\approx 1.618\), a number that recurs astonishingly often in nature, art, and architecture.
8.2 Sequences
Definition: Sequence
A sequence is a function whose domain is the set of natural numbers (or a subset \(\{1, 2, \ldots, n\}\) for some fixed \(n\)). We write the values as
\[a_1,\ a_2,\ a_3,\ \ldots\]
where \(a_k\) is the value of the function at \(k\) and is called the \(k\)th term.
Example: the sequence of even natural numbers 2, 4, 6, 8, … has \(a_n=2n\). The sequence of squares 1, 4, 9, 16, 25, … has \(a_n=n^2\).
Finite vs. infinite
A sequence with a fixed finite number of terms (say, \(n\)) is a finite sequence. One that continues indefinitely is an infinite sequence. The number of ancestors of a person, going back \(n\) generations, is finite (\(2^n\) at generation \(n\)); the sequence \(1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \ldots\) is infinite.
Specifying a sequence
A sequence can be specified in two main ways:
- Explicit formula: a closed-form rule \(a_n=f(n)\). E.g. \(a_n=n^2\) gives 1, 4, 9, 16, … directly.
- Recurrence relation: earlier terms determine later ones, with stipulated starting values. E.g. \(a_1=1\), \(a_n=a_{n-1}+2\) for \(n\ge 2\) gives 1, 3, 5, 7, 9, … .
The Fibonacci sequence
Fibonacci recurrence
\[a_1=a_2=1,\qquad a_n=a_{n-1}+a_{n-2}\ \text{for}\ n\ge 3.\]
Generates 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … . Each term is the sum of the previous two.
Series
Definition: Series
Given a sequence \(a_1, a_2, a_3, \ldots, a_n\), the expression
\[a_1+a_2+a_3+\cdots+a_n\]
is called the series associated with this sequence. A series is finite or infinite depending on whether the sequence is. Compactly, using sigma notation?:
\[\sum_{k=1}^{n}a_k=a_1+a_2+\cdots+a_n.\]
Interactive: Build Any Sequence by Formula
Choose how many terms you want and the rule. Try a few: \(a_n=n^2\), \(a_n=2n+1\), \(a_n=2^n\), \(a_n=1/n\). The first \(n\) terms and the partial sum appear instantly.
10
Terms: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Sum (Σ a_k from k=1 to 10) = 385
Sum (Σ a_k from k=1 to 10) = 385
Worked Examples
Example 1. Write the first three terms of the sequence \(a_n=\dfrac{n-3}{2(n+2)}\). Hence find the 8th term.
\(a_1=(1-3)/(2\cdot 3)=-1/3\), \(a_2=(2-3)/(2\cdot 4)=-1/8\), \(a_3=(3-3)/(2\cdot 5)=0\). Eighth term: \(a_8=(8-3)/(2\cdot 10)=5/20=1/4\).
Example 2. The 20th term of the sequence \(a_n=(n-1)(2-n)(3+n)\) equals…?
\(a_{20}=(19)(-18)(23)=-7866\).
Example 3. \(a_1=1,\ a_n=a_{n-1}+2\) for \(n\ge 2\). Find the first five terms and the corresponding series.
\(a_1=1,\ a_2=3,\ a_3=5,\ a_4=7,\ a_5=9\) (odd numbers). Series: \(1+3+5+7+9+\cdots\). (Sum of first 5 odd numbers \(=25=5^2\), a beautiful pattern!)
Example 4 (own). Compute the sum of the first 6 Fibonacci numbers.
1, 1, 2, 3, 5, 8 → sum = 20. (General fact: \(F_1+F_2+\cdots+F_n=F_{n+2}-1\); here \(F_8-1=21-1=20\) ✓.)
Activity: Find the Pattern
L3 ApplyMaterials: Calculator, paper.
Predict: What is the next term in 2, 5, 10, 17, 26, …? Can you write a formula?
- List the differences: 5−2 = 3, 10−5 = 5, 17−10 = 7, 26−17 = 9. Differences themselves form an arithmetic sequence: 3, 5, 7, 9, …
- Conjecture: the original sequence's \(n\)-th term grows like \(n^2\). Check: \(n^2+1\) gives 2, 5, 10, 17, 26 ✓. So \(a_n=n^2+1\) and \(a_6=37\).
- Now try: 1, 4, 13, 40, 121, … (each term is one more than 3× the previous). Verify \(a_n=(3^n-1)/2\).
- Try the Fibonacci ratios: \(F_2/F_1=1\), \(F_3/F_2=2\), \(F_4/F_3=1.5\), \(F_5/F_4=1.667\), \(F_6/F_5=1.6\), … . They oscillate towards a limit. What number?
The Fibonacci ratios converge to \(\varphi=(1+\sqrt 5)/2\approx 1.61803\), the golden ratio. Reason: if \(F_{n+1}/F_n\to L\), divide the recurrence \(F_{n+1}=F_n+F_{n-1}\) by \(F_n\): \(L=1+1/L\), so \(L^2-L-1=0\), giving \(L=(1+\sqrt 5)/2\). The same ratio appears in pentagons, sunflower spirals, and Renaissance art.
Competency-Based Questions
Scenario: A bacteria culture in a Petri dish doubles every hour. Starting count: 100 cells.
Q1. Write the explicit formula \(a_n\) for the count after \(n\) hours.
L3 ApplyAnswer: \(a_n=100\cdot 2^n\) (with \(a_0=100\)). After 3 hours: 800; after 5 hours: 3200.
Q2. Write the recurrence relation for the count.
L3 ApplyAnswer: \(a_0=100,\ a_n=2\,a_{n-1}\) for \(n\ge 1\).
Q3. (T/F) "Every infinite sequence has an explicit formula in elementary functions." Justify.
L5 EvaluateFalse. Many sequences (e.g. the prime numbers 2, 3, 5, 7, 11, 13, …) have no closed-form elementary formula. Recurrences and case-by-case definitions are needed.
Q4. (Fill in) The 10th Fibonacci number is ____ . Use the recurrence.
L2 UnderstandAnswer: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. So \(F_{10}=55\).
Q5. Design: a town's population doubles every 25 years. If the population is 50,000 today, write the formula for population \(P(t)\) at year \(t\) measured in years from now. When does it reach 1 million?
L6 CreateSolution: \(P(t)=50000\cdot 2^{t/25}\). Set \(P(t)=10^6\): \(2^{t/25}=20\), so \(t/25=\log_2 20\approx 4.32\), giving \(t\approx 108\) years.
Assertion–Reason Questions
Assertion (A): A sequence is a function whose domain is the natural numbers.
Reason (R): The notation \(a_1, a_2, a_3, \ldots\) shows that each natural-number index gives a unique term.
Reason (R): The notation \(a_1, a_2, a_3, \ldots\) shows that each natural-number index gives a unique term.
Answer: (a). R is the function-on-naturals view in informal notation.
Assertion (A): 1, 1, 2, 3, 5 is the start of the Fibonacci sequence.
Reason (R): Each term equals the sum of the previous two, with \(a_1=a_2=1\).
Reason (R): Each term equals the sum of the previous two, with \(a_1=a_2=1\).
Answer: (a). R is the recurrence; A lists its first 5 outputs.
Assertion (A): A sequence and a series are the same object.
Reason (R): Both involve listing numbers a₁, a₂, a₃, ….
Reason (R): Both involve listing numbers a₁, a₂, a₃, ….
Answer: (d). A is false: a sequence is a list; a series is the SUM of the list. R describes only the list aspect (and is true).
Frequently Asked Questions
What is a sequence in mathematics?
A sequence is an ordered list of numbers a₁, a₂, a₃, … . Formally, it is a function whose domain is the set of natural numbers. The kth element a_k is the kth term.
What is the difference between a sequence and a series?
A sequence is a list a₁, a₂, a₃, …. A series is the SUM of a sequence's terms: a₁ + a₂ + a₃ + …. Different objects: one is a list, the other is a sum.
What is a finite sequence?
A sequence with a finite number of terms — for example a₁, a₂, …, aₙ for some fixed n. An infinite sequence has terms continuing indefinitely.
What is the Fibonacci sequence?
The Fibonacci sequence is defined by a₁ = a₂ = 1 and aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. The first terms are 1, 1, 2, 3, 5, 8, 13, 21, 34, ….
How do you find the nth term of a sequence?
Either an explicit formula like aₙ = n(n+2) or a recurrence like aₙ = aₙ₋₁ + 2 with a₁ = 1. Substitute the index n into the formula (or apply the recurrence repeatedly) to compute any term.
What is sigma notation for series?
The sum a₁ + a₂ + … + aₙ is written compactly as Σ aₖ from k=1 to n. The symbol Σ is the Greek capital sigma; it stands for 'sum'.
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