This lesson introduces a new type of sequence, the Geometric Progression (GP). It's important to clearly distinguish it from an Arithmetic Progression (AP) and to master the two new formulas for the nth term and the sum of terms.

1. Key Concepts: What is a Geometric Progression?

A geometric progression is a sequence where you get from one term to the next by multiplying by a constant value.

• First Term (a): The starting value of the sequence.

• Common Ratio (r): The constant value you multiply by to get the next term.

  • To find 'r', divide any term by the term before it: r = T₂ / T₁.

  • 'r' can be positive, negative, a whole number, or a fraction.

How is this different from an Arithmetic Progression (AP)?

• AP: You add or subtract a common difference (d). (e.g., 2, 5, 8, 11... where d=3)

• GP: You multiply or divide by a common ratio (r). (e.g., 2, 6, 18, 54... where r=3)

2. The Core Formulas (Memorize These!)

You must know these two formulas perfectly.

Formula 1: The nth Term (Tₙ)

Tₙ = arⁿ⁻¹

• a = First term

• r = Common ratio

• n = The position of the term you want to find (e.g., for the 5th term, n=5)

Common Mistake: Using n instead of n-1 in the power. It's always "n minus one".

Formula 2: The Sum of the First n Terms (Sₙ)

There are two versions of this formula. They give the same answer, but choosing the right one makes the calculation easier.

Use when |r| > 1 (e.g., r = 2, 3, -4): Sₙ = a(rⁿ - 1) / (r - 1)

Use when |r| < 1 (e.g., r = 1/2, -1/3): Sₙ = a(1 - rⁿ) / (1 - r)

Using the correct version helps you avoid working with negative numbers in the denominator.

How to Solve Typical Exam Problems

1. Finding a Specific Term (e.g., "Find the 6th term"):

   • Identify a, r, and n.

   • Substitute these values directly into the Tₙ = arⁿ⁻¹ formula.

2. Finding the Number of Terms (n) (e.g., "Which term is 128?"):

   • You will be given Tₙ, a, and r.

   • Substitute them into Tₙ = arⁿ⁻¹.

   • This will create an equation involving indices. You will need to use your knowledge of indices to solve for n.

   • Example: If you get 2ⁿ⁻¹ = 64, you rewrite it as 2ⁿ⁻¹ = 2⁶, which means n-1 = 6, so n = 7.

3. Finding 'a' or 'r' when Two Terms are Given (e.g., "The 3rd term is 48 and the 6th is 3072"):

   • Set up two simultaneous equations using the Tₙ formula.

     • T₃ = ar² = 48 --- (1)

     • T₆ = ar⁵ = 3072 --- (2)

   • The Trick: Divide equation (2) by equation (1) to eliminate a.

     • (ar⁵) / (ar²) = 3072 / 48

     • r³ = 64

     • r = 4

   • Substitute r=4 back into equation (1) to find a.

Exam Tips & Common Pitfalls

• Tip 1: Check if it's a GP first. If the question doesn't state that the sequence is a GP, quickly check by dividing the second term by the first, and the third by the second. If the results are the same, it's a GP.

• Pitfall 1: Negative 'r'. When the common ratio 'r' is negative, always use brackets during calculations. For example, if a=5 and r=-2, the third term is 5 × (-2)² = 5 × 4 = 20, not 5 × -2² = -20.

• Pitfall 2: rⁿ vs. (r)ⁿ. Be very careful with your calculator. a × rⁿ is different from (ar)ⁿ. The power applies only to r.

• Tip 2: Two Possible Solutions. If you solve for r and get an equation like r⁴ = 81, remember that both r = 3 and r = -3 are possible solutions. This means there could be two different geometric progressions that fit the question's criteria. Read the question to see if it specifies a "positive common ratio".