Progressive Sequence Curriculum

Number Series Training

From foundational 1st differences to interleaved dual ladders, geometric ratios, Fibonacci recurrence, and quadratic polynomials ($a n^2 + b n + c$). Master the exact formula and the method of taking differences to calculate missing terms systematically.

Sequential Learning Ladder: Gradual Increased Complexity 5 Core Progression Levels

Arithmetic (1st Diff)

→

Zipped Sequences

→

Geometric (Ratio)

→

Recursive (Fibonacci)

→

Polynomial ($a n^2+b n+c$)

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Level 1: Core Additive Not Started

Simple Arithmetic Sequence (Constant 1st Difference)

$x_n = x_1 + (n - 1)d \quad \iff \quad x_n = x_{n-1} + d$

In an arithmetic sequence, the difference between consecutive terms is constant ($d = x_n - x_{n-1}$). To find the next number, calculate the first difference $\Delta^{(1)}$ and add it to the last term.

Study Lesson Practice (10 Q)

Level 2: Dual Ladder Not Started

Two Arithmetic Sequences Zipped (Interleaved / Alternating)

$x_{2k-1} = x_1 + (k-1)d_1 \quad \text{and} \quad x_{2k} = x_2 + (k-1)d_2$

Two independent arithmetic sequences are interwoven into alternate odd and even positions. Look at every second number (stride-2) or detect alternating jump deltas (+d₁, +d₂).

Study Lesson Practice (10 Q)

Level 3: Multiplicative Not Started

Geometric Sequence (Constant Ratio)

$x_n = x_1 \cdot r^{n-1} \quad \iff \quad x_n = x_{n-1} \cdot r \quad \left(r = \frac{x_n}{x_{n-1}}\right)$

In a geometric sequence, each term is multiplied by a constant ratio $r$. When adjacent differences grow exponentially rather than staying steady, test division instead of subtraction.

Study Lesson Practice (10 Q)

Level 4: Recurrence Not Started

Recursive Sequence (Fibonacci & Linear Recurrence)

$x_n = x_{n-1} + x_{n-2} \quad \text{or} \quad x_n = a \cdot x_{n-1} + b \cdot x_{n-2}$

In a recursive sequence, each term depends directly on the preceding two terms. In Fibonacci-type series, every number is the exact sum of the two numbers before it.

Study Lesson Practice (10 Q)

Level 5: 2nd Differences Not Started

Polynomial Sequences & Method of Differences (a·n² + b·n + c)

$x_n = a n^2 + b n + c \iff \Delta^{(2)} = 2a \quad \text{(Constant 2nd Difference)}$

When 1st differences are not constant, take the differences of the differences (the 2nd differences). For any quadratic sequence, the 2nd differences are CONSTANT ($2a$). Extend the 2nd difference line to deduce the next number.

Study Lesson Practice (10 Q)

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