What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant number.

• This constant is called the common ratio and is usually denoted by r.

• Each term can be found by multiplying the previous term by r.

Example:

• Here, the common ratio r = 2 (4 ÷ 2 = 2, 8 ÷ 4 = 2, …)

• This is a geometric sequence.

Step 1: Identify the first term and common ratio

• First term: a₁

• Common ratio: r = a₂ ÷ a₁

Example:

• a₁ = 3

• r = 6 ÷ 3 = 2

Step 2: Find the nth term

The nth term formula for a geometric sequence is:

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}an​=a1​⋅rn−1

Where:

• a_n = nth term

• a₁ = first term

• r = common ratio

• n = term number

Example:

Find the 5th term:

a5=3⋅25−1=3⋅24=3⋅16=48a_5 = 3 \cdot 2^{5-1} = 3 \cdot 2^4 = 3 \cdot 16 = 48a5​=3⋅25−1=3⋅24=3⋅16=48

✅ 5th term = 48

Step 3: Find the sum of n terms

The sum of the first n terms of a geometric sequence depends on whether r = 1 or r ≠ 1.

If r ≠ 1:

Sn=a11−rn1−rorSn=a1rn−1r−1S_n = a_1 \frac{1 – r^n}{1 – r} \quad \text{or} \quad S_n = a_1 \frac{r^n – 1}{r – 1}Sn​=a1​1−r1−rn​orSn​=a1​r−1rn−1​

Example:

S4=234−13−1=281−12=2⋅40=80S_4 = 2 \frac{3^4 – 1}{3 – 1} = 2 \frac{81 – 1}{2} = 2 \cdot 40 = 80S4​=23−134−1​=2281−1​=2⋅40=80

✅ Sum of first 4 terms = 80

If r = 1:

• All terms are the same, so sum = n * a₁

Step 4: Write a general geometric sequence

• General term formula: a_n = a₁ * r^(n-1)

• Sequence example: a₁ = 5, r = 2

✅ Sequence: 5, 10, 20, 40, 80, …

Step 5: Real-life examples

Common beginner mistakes

1. ❌ Forgetting to divide terms to find r

2. ❌ Using addition instead of multiplication

3. ❌ Using the wrong formula for sum

4. ❌ Confusing geometric and arithmetic sequences

Summary

• Geometric sequence = multiply by constant ratio r

• nth term: a_n = a₁ * r^(n-1)

• Sum of first n terms: S_n = a₁ (r^n - 1)/(r - 1) if r ≠ 1

• Useful for growth, decay, investments, and doubling/halving patterns