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Harit Learning

Arithmetic & Geometric Sequences Lab

Module 1: Introduction to Sequences

What is a Sequence?

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term.

Arithmetic Sequence (AP)

A sequence where the difference between consecutive terms is constant.

Example: 2, 5, 8, 11, ... (+3 each time)

Geometric Sequence (GP)

A sequence where the ratio between consecutive terms is constant.

Example: 3, 6, 12, 24, ... (x2 each time)

Interactive Sequence Generator

Enter a starting term and a rule to generate your own sequence.

Module 2: Arithmetic Sequence (AP)

Understanding AP

An Arithmetic Sequence adds or subtracts a constant value each time.

  • First Term: a
  • Common Difference: d = t2 - t1
  • Number of terms: n
General Term: an = a + (n - 1)d

Solved Example:

Find the 10th term of 2, 5, 8...

a = 2, d = 3, n = 10
a10 = 2 + (10-1)3 = 2 + 27 = 29

Interactive AP Explorer

a_5 = 2 + (5-1)3 = 14

Module 3: Geometric Sequence (GP)

Understanding GP

A Geometric Sequence multiplies or divides by a constant value each time.

  • First Term: a
  • Common Ratio: r = t2 / t1
General Term: an = a × r(n - 1)

Solved Example:

Find the 6th term of 3, 6, 12...

a = 3, r = 2, n = 6
a6 = 3 × 2(6-1) = 3 × 32 = 96

Interactive GP Explorer

a_6 = 3 × 2^(5) = 96

Module 4: Arithmetic Mean (AM)

The Arithmetic Mean between two numbers a and b is the number that, when placed between them, forms an Arithmetic Sequence.

AM = (a + b) / 2

AM Calculator & Visualizer

6 ➔ 10 ➔ 14

Sequence: 6, 10, 14 (Common diff: 4)

Module 5: Geometric Mean (GM)

The Geometric Mean between two positive numbers a and b creates a Geometric Sequence.

GM = √(a × b)

GM Calculator & Visualizer

4 ➔ 6 ➔ 9

Sequence: 4, 6, 9 (Common ratio: 1.5)

Module 6: Sum of Arithmetic Sequence

The sum of the first n terms of an AP is denoted by Sn.

Sn = n/2 [2a + (n - 1)d]

Or if the last term (l) is known: Sn = n/2 (a + l)

Interactive Sum Calculator

Sum = 25

Module 7: Sum of Geometric Sequence

Formulas depend on the value of the common ratio r:

  • If r > 1: Sn = a(rn - 1) / (r - 1)
  • If r < 1: Sn = a(1 - rn) / (1 - r)
  • Infinite Sum (only if |r| < 1): S∞ = a / (1 - r)

GP Sum Calculator & Infinite Limit

Sum = 15

Module 8: Numerical Problem Solver

Click below to generate a random numerical problem and solve it step-by-step.

Module 9: Verbal Word Problems

Sequences appear everywhere in real life: finance, biology, sports, and business.

Module 10: Graph Visualization Lab

Compare the linear growth of an Arithmetic Sequence vs the exponential growth/decay of a Geometric Sequence.

Arithmetic Settings

Geometric Settings

■ AP (Linear) ■ GP (Exponential)

Module 11: Formula Reference Section

Arithmetic Sequence (AP)

  • General Term:
    an = a + (n - 1)d
  • Common Difference:
    d = an - an-1
  • Arithmetic Mean:
    AM = (a + b) / 2
  • Sum (n terms):
    Sn = n/2 [2a + (n - 1)d]
  • Sum (first & last):
    Sn = n/2 (a + l)

Geometric Sequence (GP)

  • General Term:
    an = arn-1
  • Common Ratio:
    r = an / an-1
  • Geometric Mean:
    GM = √(ab)
  • Sum (r > 1):
    Sn = a(rn - 1) / (r - 1)
  • Sum (r < 1):
    Sn = a(1 - rn) / (1 - r)
  • Infinite Sum (|r|<1):
    S∞ = a / (1 - r)

Module 12: Interactive Quiz System

Test Your Knowledge!

30 High-Quality Questions covering AP, GP, Means, Sums, and Word Problems.

  • ✔️ Multiple Choice
  • ✔️ Instant Feedback
  • ✔️ Performance Grading

Module 13: Unlimited Practice Mode

Module 14: Teacher & Student Dashboard

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Student Performance Report

Harit Learning Sequences Lab

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