Predicting What Comes Next: Exploring Sequences and Progressions (Class 9): Notes और MCQ Practice

आसान भाषा में

Sequence संख्याओं की एक ordered list होती है जो किसी खास pattern या rule पर चलती है। अगर consecutive terms के बीच का difference हमेशा constant रहे, तो उसे Arithmetic Progression (AP) कहते हैं, और अगर terms को निकालने के लिए हर बार constant number से multiply किया जाए (constant ratio), तो उसे Geometric Progression (GP) कहते हैं!

Important Points & Formulas

• Notation: Sequence का $n$-th term $t_n$, $s_n$ या $u_n$ से दर्शाया जाता है, जहाँ $n$ हमेशा एक Natural Number (position) होता है।
• Explicit vs Recursive Rule: Explicit formula direct term देता है (e.g., $u_n = 2n - 1$). Recursive formula अगले term को पिछले term से जोड़ता है (e.g., $t_n = t_{n-1} + d$).
• Virahānka–Fibonacci Sequence: $1, 2, 3, 5, 8, ...$ जहाँ $V_1 = 1, V_2 = 2$ और $V_n = V_{n-1} + V_{n-2}$ ($n ≥ 3$).
• Arithmetic Progression (AP):
• $n$-th term: $t_n = a + (n - 1) × d$ (जहाँ $a$ = first term, $d$ = common difference).
• Recursive rule: $t_1 = a, t_n = t_{n-1} + d$.
• Ordered pairs $(n, t_n)$ graph पर हमेशा एक straight line बनाते हैं।
• Sum of First $n$ Natural Numbers (Āryabhaṭīya rule):
• $S_n = n(n + 1) ÷ 2$
• $n$-th Triangular Number का formula भी यही होता है: $t_n = n(n + 1) ÷ 2$.
• Geometric Progression (GP):
• $n$-th term: $t_n = a × r^{n-1}$ (जहाँ $a$ = first term, $r$ = common ratio).

2 Memory Tricks

1. AP मतलब 'Addition', GP मतलब 'Product': AP में हर step पर common difference $d$ add होता है, जबकि GP में common ratio $r$ multiply होता है।
2. (n − 1) Rule: First term में कोई change नहीं होता, इसलिए $n$-th term तक पहुँचने के लिए केवल $(n - 1)$ बार $d$ जोड़ा जाता है या $r$ की power $(n - 1)$ चढ़ाई जाती है।

2 Common Board Exam Mistakes

1. $n$ का natural number न होना भूल जाना: जब check करना हो कि कोई number term है या नहीं, तो equation solve करने पर यदि $n$ fraction या decimal आता है (जैसे $94.6$), तो वह sequence का term नहीं हो सकता।
2. GP formula में power $n$ लिख देना: विद्यार्थी अक्सर $t_n = a × r^n$ लिख देते हैं, जबकि सही formula $t_n = a × r^{n-1}$ होता है।

Predicting What Comes Next: Exploring Sequences and Progressions

1. Chapter in Short

This chapter introduces the concept of number patterns and sequences. A sequence is an ordered list of numbers where each number is called a term. We learn two main ways to describe sequences: using an explicit rule (calculating any term directly using its position number $n$) and a recursive rule (finding a term using one or more preceding terms).

The chapter explores two special types of progressions:

• Arithmetic Progressions (APs): Sequences where consecutive terms differ by a constant value called the common difference.
• Geometric Progressions (GPs): Sequences where consecutive terms have a constant multiplier called the common ratio.

It also covers the formula for the sum of the first $n$ natural numbers, its visual representation, and historical contributions from Indian mathematicians like Virahānka and Āryabhaṭa.

2. Key Concepts Explained Simply

Sequences and Terms

• A sequence is an ordered list of numbers following a specific order or pattern.
• Each individual number in the list is called a term.
• We label terms by their position: $t₁$ is the 1st term, $t₂$ is the 2nd term, and $t_n$ is the $n$-th term.
• Finite Sequence: A sequence with a limited number of terms (e.g., $6, 12, 24, 48, 96$).
• Infinite Sequence: A sequence that continues without end, indicated by three dots "..." (e.g., Natural numbers: $1, 2, 3, 4, ...$).

Special Sequences

• Triangular Numbers: $1, 3, 6, 10, 15, 21, ...$

Each term is the sum of natural numbers up to that term:

$1 = 1$

$3 = 1 + 2$

$6 = 1 + 2 + 3$

$10 = 1 + 2 + 3 + 4$

• Square Numbers: $1, 4, 9, 16, 25, 36, ...$

Each term is the sum of odd numbers up to that term:

$1 = 1$

$4 = 1 + 3$

$9 = 1 + 3 + 5$

$16 = 1 + 3 + 5 + 7$

Explicit Rule vs. Recursive Rule

• Explicit Rule: A formula where you plug in the position number $n$ directly to find the term without needing to know any other terms.
• Example: For odd numbers, $u_n = 2n - 1$. To find the 50th term, substitute $n = 50$: $u₅₀ = 2(50) - 1 = 99$.
• Recursive Rule: A rule that gives the first term(s) and tells you how to get the next term from the term(s) before it.
• Example: $t₁ = 1, t_n = t_{n-1} + 3$ for $n ≥ 2$. You must know the previous term to find the next.

Arithmetic Progressions (AP)

• An AP is a sequence where the difference between any term and its previous term is always the same constant value, called the common difference ($d$).
• If the first term is $a$ and the common difference is $d$:
• The sequence looks like: $a, a + d, a + 2d, a + 3d, ...$
• The $n$-th term is: $t_n = a + (n - 1) × d$
• The recursive rule is: $t₁ = a, t_n = t_{n-1} + d$ for $n ≥ 2$
• Everyday Example: A taxi ride charging a fixed booking fee of ₹200 plus ₹40 per km. The fare sequence for 1 km, 2 km, 3 km... is ₹240, ₹280, ₹320... Here, $a = 240$ and $d = 40$.
• Graphing an AP: When the pairs (position $n$, value $t_n$) are plotted on a graph, all points lie on a straight line.

Geometric Progressions (GP)

• A GP is a sequence where each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio ($r$).
• Ratio of consecutive terms is constant: $t₂ ÷ t₁ = t₃ ÷ t₂ = ... = r$
• The sequence looks like: $a, a × r, a × r², a × r³, ...$
• The $n$-th term is: $t_n = a × r^{(n-1)}$
• The recursive rule is: $t₁ = a, t_n = r × t_{n-1}$ for $n ≥ 2$
• Example: Green square patterns doubling at each stage: $3, 6, 12, 24, 48...$ where $a = 3$ and $r = 2$.

3. Important Definitions, Formulas, Facts, and Dates

• Sequence: An ordered list of numbers where each number is a term.
• Term Position ($n$): In the notation $t_n$, $n$ is always a positive integer (natural number: $1, 2, 3, ...$), though the value of $t_n$ itself can be negative, zero, a fraction, or any real number.
• Explicit Formula of an AP:

$t_n = a + (n - 1) × d$

where $a$ is the first term and $d$ is the common difference ($d = t_n - t_{n-1}$).

• Recursive Formula of an AP:

$t₁ = a$, and $t_n = t_{n-1} + d$ for $n ≥ 2$.

• Explicit Formula of a GP:

$t_n = a × r^{(n-1)}$

where $a$ is the first term and $r$ is the common ratio ($r = t_n ÷ t_{n-1}$).

• Sum of the First $n$ Natural Numbers ($S_n$):

$S_n = 1 + 2 + 3 + ... + n = n(n + 1) ÷ 2$

• $n$-th Triangular Number:

$t_n = n(n + 1) ÷ 2$

• Āryabhaṭa's Rule (Āryabhaṭīya, Chapter 2, Verse 19):

The sum of natural numbers is calculated by taking the average of the first and last terms $[(1 + n) ÷ 2]$ and multiplying by the number of terms ($n$).

• Virahānka–Fibonacci Sequence:

$V₁ = 1, V₂ = 2$, and $V_n = V_{n-1} + V_{n-2}$ for $n ≥ 3$.

The sequence is: $1, 2, 3, 5, 8, 13, 21, 34, ...$

• First written down by Indian scholar Virahānka in Vṛttajātisamuchaya (7th century CE) while analyzing poetic meter.
• Later studied by Indian scholars Gopāla (c. 1135 CE) and Hemachandra (c. 1150 CE).
• Studied later in Europe by Fibonacci (c. 1200 CE).

4. Solved Examples

Example 1: Finding Terms and Checking Belonging in an AP

Question: Consider the AP: $21, 18, 15, ...$

(i) Find the 15th term.

(ii) Which term of this AP is $-81$?

(iii) Is $0$ a term of this AP?

Solution:

• Here, first term $a = 21$.
• Common difference $d = 18 - 21 = -3$.

(i) Formula: $t_n = a + (n - 1) × d$

$t₁₅ = 21 + (15 - 1) × (-3)$

$t₁₅ = 21 + 14 × (-3) = 21 - 42 = -21$

(ii) Set $t_n = -81$:

$21 + (n - 1) × (-3) = -81$

$(n - 1) × (-3) = -81 - 21 = -102$

$n - 1 = -102 ÷ (-3) = 34$

$n = 34 + 1 = 35$

Thus, $-81$ is the 35th term of the AP.

(iii) Set $t_n = 0$:

$21 + (n - 1) × (-3) = 0$

$(n - 1) × (-3) = -21$

$n - 1 = -21 ÷ (-3) = 7$

$n = 7 + 1 = 8$

Since $n = 8$ is a natural number, $0$ is indeed the 8th term of the sequence.

Example 2: Finding a Specific Term from Two Given Terms

Question: An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term.

Solution:

• Let the first term be $a$ and the common difference be $d$.
• 3rd term: $t₃ = a + (3 - 1)d = a + 2d = 12$ --- (Equation 1)
• Total terms = 50, so the last term is the 50th term:

$t₅₀ = a + (50 - 1)d = a + 49d = 106$ --- (Equation 2)

Subtract Equation 1 from Equation 2:

$(a + 49d) - (a + 2d) = 106 - 12$

$47d = 94$

$d = 94 ÷ 47 = 2$

Substitute $d = 2$ in Equation 1:

$a + 2(2) = 12$

$a + 4 = 12$

$a = 8$

Now find the 29th term:

$t₂₉ = a + (29 - 1)d = 8 + 28 × 2 = 8 + 56 = 64$

Answer: The 29th term is 64.

Example 3: Sum of Natural Numbers Between Limits

Question: Find the sum of consecutive numbers: $25 + 26 + 27 + ... + 58$.

Solution:

We know the sum of the first $n$ natural numbers is $S_n = n(n + 1) ÷ 2$.

We can write:

$25 + 26 + ... + 58 = (1 + 2 + ... + 58) - (1 + 2 + ... + 24)$

$= S₅₈ - S₂₄$

Calculate $S₅₈$:

$S₅₈ = (58 × 59) ÷ 2 = 29 × 59 = 1711$

Calculate $S₂₄$:

$S₂₄ = (24 × 25) ÷ 2 = 12 × 25 = 300$

Subtract the two sums:

Sum $= 1711 - 300 = 1411$

Answer: The sum is 1411.

Example 4: Real-Life Application of an AP

Question: Harish started work at an annual salary of ₹5,00,000 and received an increment of ₹20,000 each year. After how many years did his income reach ₹7,00,000?

Solution:

• Annual salaries form an AP where:
• Initial salary ($t₁$): $a = 5,00,000$
• Annual increment: $d = 20,000$
• Target salary: $t_n = 7,00,000$

Using $t_n = a + (n - 1)d$:

$5,00,000 + (n - 1) × 20,000 = 7,00,000$

$(n - 1) × 20,000 = 7,00,000 - 5,00,000 = 2,00,000$

$n - 1 = 2,00,000 ÷ 20,000 = 10$

$n = 11$

So, his income reaches ₹7,00,000 in the 11th year, which is after 10 years of service.

Answer: After 10 years.

5. Common Mistakes and Quick Revision Points

• Checking if a number belongs to a sequence: Always solve for $n$. If $n$ comes out as a fraction or decimal (e.g., $94.6$) or a negative number, the given number cannot be a term of the sequence, because positions are strictly natural numbers ($1, 2, 3, ...$).
• AP Common Difference Sign: For decreasing APs (e.g., $11, 7, 3, -1...$), the common difference is negative: $d = 7 - 11 = -4$. Do not forget the negative sign.
• First Term vs. Zero-th Term: Remember that the first term is $t₁$, not $t₀$. In an AP, the $n$-th term is $a + (n - 1)d$, not $a + nd$.
• Summing a Range ($a$ to $b$): To find $25 + 26 + ... + 58$, subtract $S₂₄$, not $S₂₅$, from $S₅₈$. The term 25 must be included in the sum!
• AP vs. GP Difference:
• In an AP, consecutive terms have a common difference: $t_n - t_{n-1} = d$.
• In a GP, consecutive terms have a common ratio: $t_n ÷ t_{n-1} = r$.