How to Solve Geometric Progression Problems
Solve geometric progression aptitude problems using the nth-term and sum formulas, plus the sum-to-infinity case, with worked examples and practice.
Expected Interview Answer
A geometric progression (GP) is a sequence where each term is the previous term multiplied by a fixed common ratio r, so the nth term is a×r^(n-1) and the sum of n terms is a(r^n − 1)/(r − 1) for r ≠ 1.
Once you identify a and r from two known terms, every other term or partial sum follows directly from the formulas. For |r| < 1, the sum to infinity converges to a/(1 − r), a distinct case from the finite-sum formula and a frequent trap when r is not checked first. Ratios can be found by dividing any term by its predecessor, and three consecutive GP terms satisfy the property that the middle term squared equals the product of its neighbors. Always verify r is constant across the given terms before applying any formula, since a sequence that merely looks geometric may not be.
- One formula pair (nth term, sum) solves almost every GP question
- The infinite-sum case unlocks recurring-decimal and limit problems
- The middle-term-squared property quickly verifies three-term GPs
AI Mentor Explanation
A batter whose strike rate doubles every over — 4 runs, then 8, then 16, then 32 — is scoring in a geometric progression with common ratio 2, not adding a fixed amount like arithmetic progression would. Predicting the runs in the 6th over just means multiplying the first over’s runs by 2 raised to the power of 5. If the innings kept doubling forever in a shrinking-ratio sense, commentators could even quote a theoretical total using the sum-to-infinity idea, though real cricket always has a finite number of overs.
Worked example
First term & ratio
- a = 3, r = 2
6th term
- T6 = 3×2^5 = 96
Sum of 6 terms
- S6 = 3(2^6−1)/(2−1) = 189
Step-by-Step Explanation
Step 1
Identify a and r
Divide any term by its predecessor to confirm a constant common ratio r.
Step 2
Apply the nth-term formula
Tn = a × r^(n-1) gives any specific term.
Step 3
Apply the sum formula
Sn = a(r^n − 1)/(r − 1) for r ≠ 1 gives the sum of n terms.
Step 4
Check for infinite sum
If |r| < 1 and the series is infinite, use S∞ = a/(1 − r) instead.
What Interviewer Expects
- Correct identification of a and r from given terms
- Correct nth-term and finite-sum formulas
- Knowing when to switch to the sum-to-infinity formula
- Verifying r is constant before applying any GP formula
Common Mistakes
- Using the arithmetic progression sum formula on a geometric sequence
- Applying the infinite-sum formula when |r| ≥ 1, where it does not converge
- Sign errors when r is negative, producing an alternating sequence
- Confusing the common ratio with the common difference
Best Answer (HR Friendly)
“A geometric progression multiplies by a fixed ratio r each step rather than adding a fixed amount. Once you know the first term and r, the nth term is a times r to the power n minus 1, and the sum of n terms follows a clean formula. If the ratio’s size is less than 1 and the series never ends, the sum settles at a finite limit given by a over 1 minus r — that is the sum-to-infinity case.”
Follow-up Questions
- How do you find the sum to infinity of a geometric series and when does it apply?
- How would you insert geometric means between two numbers?
- How does a negative common ratio change the sequence behavior?
- How do you convert a recurring decimal into a fraction using GP sum-to-infinity?
MCQ Practice
1. A GP has first term 5 and common ratio 3. What is the 4th term?
T4 = 5×3^3 = 5×27 = 135.
2. Find the sum to infinity of the GP 8, 4, 2, 1, ...
a = 8, r = 1/2. S∞ = 8/(1 − 0.5) = 16.
3. Three numbers x, 6, y form a GP. If x = 2, what is y?
Middle term squared equals product of neighbors: 6^2 = 2×y → 36 = 2y → y = 18.
Flash Cards
GP nth-term formula? — Tn = a × r^(n-1).
GP finite-sum formula? — Sn = a(r^n − 1)/(r − 1), for r ≠ 1.
GP sum-to-infinity formula? — S∞ = a/(1 − r), valid only when |r| < 1.
Property of 3 consecutive GP terms? — Middle term squared = product of the two neighbors.