What is a Geometric Progression?
A Geometric Progression (GP) is a sequence in which each term is obtained by multiplying the previous term by a constant value called the Common Ratio ($r$).
The Bacteria Growth & Multiplier Model
Imagine a bacteria population that doubles every hour. Starting with 2 bacteria: 2, 4, 8, 16, 32... Each term is multiplied by a constant factor of 2.
Geometric Progression is an essential Modern Math topic that appears in CAT, XAT, NMAT, SNAP, and MBA CET, particularly in questions involving exponential growth, decay, compound interest, bouncing balls, and pattern recognition.
Connected Vault Modules:
Use when ratio $r > 1$. Alternatively use $\frac{a(1 - r^n)}{1 - r}$ when $r < 1$.
Valid ONLY when absolute value of common ratio is less than 1 ($|r| < 1$).
Divide any term by its immediately preceding term.
The geometric mean of two positive numbers $a$ and $b$.
Finding the 8th Term of a GP
Q: Find the 8th term of the GP: 3, 9, 27, ...
Step 1: Identify first term $a = 3$ and common ratio $r = 9 / 3 = 3$.
Step 2: Apply $a_n = a \cdot r^{n-1}$ for $n = 8$:
$$a_8 = 3 \times 3^{7} = 3^8 = \mathbf{6561}$$
Sum of a Finite GP Series
Q: Find the sum of: 5, 10, 20, 40.
Step 1: Identify $a = 5, r = 2, n = 4$.
Step 2: Apply $S_n = \frac{a(r^n - 1)}{r - 1}$:
$$S_4 = \frac{5(2^4 - 1)}{2 - 1} = \frac{5(16 - 1)}{1} = 5 \times 15 = \mathbf{75}$$
Bouncing Ball Rebound Height Problem
Q: A ball rebounds to half its previous height after every bounce. If it is dropped from a height of 32 meters, what is the height after the 5th bounce?
Step 1: Bounce heights form a GP: initial drop = 32m, 1st bounce = 16m, 2nd = 8m...
Step 2: Here $a = 32$ and common ratio $r = 1/2$.
Step 3: The 5th bounce corresponds to the 6th term of the sequence ($a_6$):
$$a_6 = 32 \times \left(\frac{1}{2}\right)^5 = 32 \times \frac{1}{32} = \mathbf{1\text{ meter}}$$
❌ Trap 1: Using AP Formulas
Adding $d$ instead of multiplying $r$
Always verify whether consecutive terms have a constant difference or a constant ratio before solving!
❌ Trap 2: Invalid Infinite Sum
Applying S_∞ when |r| ≥ 1
The infinite sum formula $S_\infty = a / (1-r)$ exists ONLY if $|r| < 1$. For $r \ge 1$, the sum is infinite!
❌ Trap 3: Off-by-One Exponent
a · r^n ≠ a_n
The $n$-th term formula uses exponent $n - 1$ ($a_n = a \cdot r^{n-1}$). For the 5th bounce from height $h$, calculate term 6!
⚡ Common Multiplier Pattern Recognition
When word problems mention "doubles every year" or "halves after every bounce", immediately identify $r=2$ or $r=1/2$ without calculation!
⚡ Infinite Series Fast Sum
For infinite decaying series $1 + 0.5 + 0.25 + ...$: $S_\infty = 1 / (1 - 0.5) = 1 / 0.5 = \mathbf{2}$.
Question 1:
Find the 10th term of the GP: 2, 6, 18, ...
Solution: $a = 2, r = 6 / 2 = 3, n = 10$.
a_{10} = 2 × 3^{9} = 2 × 19683 = 39,366.
Question 2:
Find the sum of the first 6 terms of the GP: 1, 2, 4, 8, ...
Solution: $a = 1, r = 2, n = 6$.
S_6 = \frac{1(2^6 - 1)}{2 - 1} = \frac{64 - 1}{1} = 63.
Question 3:
Find the common ratio of the sequence: 7, 21, 63, 189, ...
Solution: $r = T_2 / T_1 = 21 / 7 = 3.
Q: What is the fundamental definition of a Geometric Progression (GP)?
A Geometric Progression (GP) is a sequence in which each term is obtained by multiplying the previous term by a constant value called the Common Ratio ($r$).
Q: When does an infinite GP have a finite sum?
Only when the absolute value of the common ratio is less than 1, i.e., $|r| < 1$. The sum is given by $S_\infty = a / (1 - r)$.
Q: Can the common ratio of a GP be negative?
Yes. A negative common ratio produces alternating positive and negative terms (e.g. 3, -6, 12, -24...).