What is the rule derived for the sequence 1/28, 4/28, 9/28, ...?

The rule derived for this sequence, where the numerator follows the pattern 1, 4, 9, ..., is n² divided by 28, or $n^2 / 28$.

How was the rule formulated for the sequence 1, 9, 27, ...?

The first term (n=1) is 1, but the subsequent terms showed a pattern of multiplication by 3. The correct rule found was $3^n$, confirmed by checking the third term: $3^3 = 27$.

What is the common difference or pattern in the sequence 20, 16, 12, 8, ...?

The pattern is subtracting 4 from the preceding term. The next three terms found were 4, 0, and -4.

What is the nth term rule found for the sequence 20, 16, 12, 8, ...?

The rule derived was $24 - 4n$, which can also be expressed as $-4n + 24$.

What is the pattern observed in the sequence 1/2, 1/3, 1/4, 1/5, ...?

The numerator remains constant at 1, and the denominator follows the pattern n+1, where n is the term number. Thus, the rule is $1 / (n+1)$.

FORMULATING THE RULE IN FINDING THE NTH TERM USING DIFFERENT STRATEGIES

Formulating the nth Term Rule for Sequences
📌 A sequence is a set of numbers arranged by a definite rule; understanding this rule helps in continuing or filling missing parts of the sequence.
📌 Strategies involve identifying the common difference, ratio, or relationship between the term number ($n$) and the term value ( $a_n$ ).
📌 The $n^{th}$ term rule is often expressed in the form $a_n = dn + c$ (for arithmetic sequences) or using exponents for geometric/power sequences.

Example Sequence Analysis
📌 For the sequence 5, 10, 15, 20..., the rule is $a_n = 5n$ ; the $7^{th}$ term is $5 \times 7 = 35$ .
📌 For the sequence 3, 5, 7, 9... (common difference of 2), the rule is $a_n = 2n + 1$ ; checking the $5^{th}$ term: $2(5) + 1 = 11$.
📌 For sequences like -2, -4, -6..., the operation is subtracting 2 (or adding -2), leading to the rule $a_n = -2n$ .

Sequences Involving Exponents and Fractions
📌 For sequences where the numerator follows a pattern, like $\frac{1}{28}, \frac{4}{28}, \frac{9}{28}, \frac{16}{28}...$ , the numerator is $n^2$ , resulting in the rule $a_n = \frac{n^2}{28}$ .
📌 For the sequence 1, 3, 9, 27... (geometric progression with ratio 3), the rule is $a_n = 3^n$ ; the $5^{th}$ term is $3^5 = 243$ .

Linear Rules with Constants
📌 For the sequence 20, 16, 12, 8... (common difference is -4), the rule derived is $a_n = -4n + 24$ (or $24 - 4n$).
📌 For sequences like $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}...$ , the numerator is always 1, and the denominator is $n+1$, yielding the rule $a_n = \frac{1}{n+1}$ .

Key Points & Insights
➡️ To find the nth term rule, systematically check if the difference between consecutive terms is constant (linear rule) or if terms are multiplied/raised to a power (exponential/power rule).
➡️ When dealing with fractions, analyze the numerator and denominator sequences independently to formulate their respective rules before combining them into $a_n$ .
➡️ For sequences with a constant difference $d$, the general formula often involves $dn$; subsequent steps determine the required constant offset ($+c$ or $-c$).

📸 Video summarized with SummaryTube.com on Nov 25, 2025, 04:36 UTC

FORMULATING THE RULE IN FINDING THE NTH TERM USING DIFFERENT STRATEGIES

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