NDA Maths · Sequence & Series

Geometric Progressions — the constant-ratio engine

A list where each term is the one before it times a fixed ratio — so terms grow (or shrink) by multiplication, and an unending GP can still add up to a finite total.

All Sequence & Series notes NDA Maths strategy

Why this matters

Nineteen PYQs across 2017–2026, mostly EASY–MODERATE. NDA tests the nth term, the finite sum, and above all the infinite sum (when the ratio lies strictly between minus one and one) — repeating decimals, continued fractions, and infinite products all reduce to it. Two more recurring shapes: GP-preserving operations and the product-of-terms symmetry. Five concepts cover the lot.

Concept 1 of 5

nth term, geometric mean, and the three-term condition

Intuition

In a GP every step multiplies by the same number — the common ratio

r

. So the nth term is the first term times

r

raised to

(n-1)

. Three numbers are in GP when the middle is the geometric mean of the outer two: its square equals their product.

Definition

A geometric progression has first term $a$ and common ratio $r = \dfrac{a_{k+1}}{a_k}$ (constant, $r \ne 0$ ). Then:

nth term: $a_n = a\,r^{\,n-1}$ .
Geometric mean of $a$ and $b$ : $\text{GM} = \sqrt{ab}$ .
Three-term condition: $a, b, c$ are in GP $\iff b^2 = ac$ .

nth term and three-term condition

a_n = a\,r^{\,n-1}, \qquad b^2 = ac \ \text{(for } a,b,c \text{ in GP)}

$a$ first term
$r$ common ratio

Worked example

Find the 6th term of the GP

3, 6, 12, 24, \ldots

First term $a = 3$ ; common ratio $r = 6/3 = 2$ .
$a_6 = a\,r^{5} = 3 \times 2^{5} = 3 \times 32$ .

Answer:

a_6 = 96

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the 5th term of the GP

2, -6, 18, \ldots

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
4th term of $1, 3, 9, \ldots$ ?
2.
Geometric mean of $4$ and $9$ ?
3.
If $2, x, 8$ are in GP, find $x$ (positive value).
4.
Common ratio of $81, 27, 9, \ldots$ ?

From the bank · past-year question

Example 1Sequence & SeriesEASY

What is the

n^{th}

term of the sequence

25, -125, 625, -3125, \ldots

[Q1 · Apr · 2019]

Drill 1 more on nth term, geometric mean, and the three-term condition

Concept 2 of 5

Sum of a finite GP

Intuition

Adding up a GP has a one-line formula because almost everything cancels when you subtract

r

times the sum from the sum itself. The same formula handles repeating-digit sums (

0.3 + 0.33 + 0.333 + \cdots

) once you factor each term into a GP.

Definition

For a GP with first term $a$ , ratio $r \ne 1$ , and $n$ terms:

S_n = \frac{a\,(r^n - 1)}{r - 1} = \frac{a\,(1 - r^n)}{1 - r}.

Use the first form when

r > 1

, the second when

r < 1

(both are equal). If

r = 1

the GP is constant and

S_n = na

Sum of n terms

S_n = \frac{a\,(r^n - 1)}{r - 1}\quad (r \ne 1)

Worked example

Find the sum of the first 6 terms of the GP

2, 6, 18, \ldots

$a = 2,\ r = 3,\ n = 6$ .
$S_6 = \dfrac{2\,(3^6 - 1)}{3 - 1} = \dfrac{2\,(729 - 1)}{2} = 729 - 1$ .

Answer:

728

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the sum of the first 5 terms of

1, \tfrac12, \tfrac14, \ldots

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Sum of $1 + 2 + 4 + 8 + 16$ ?
2.
Sum of $3 + 9 + 27$ ?
3.
Sum of first 4 terms of $1, \tfrac13, \tfrac19, \ldots$ ?
4.
If $r = 1$ and $a = 5$ , $S_n = ?$

From the bank · past-year question

Example 2Sequence & SeriesMODERATE

What is the sum of the series

0.3 + 0.33 + 0.333 + \ldots

n

terms?

[Q8 · Apr · 2017]

Repeating-digit sums hide a GP

For

0.3 + 0.33 + 0.333 + \cdots

, write each term as

\tfrac{3}{9}(1 - 10^{-k})

. The sum splits into a constant part (an AP-like count of

\tfrac13

) and a true GP

\sum 10^{-k}

. Don't try to treat the original list as a GP directly — it isn't one.

Drill 3 more on sum of a finite gp

Concept 3 of 5

Sum of an infinite GP

Intuition

If the ratio is between

-1

and

1

, the terms shrink toward zero fast enough that the whole unending sum settles on a finite number. That single formula cracks repeating decimals, nested continued fractions, and infinite products (take logs first for products).

Definition

For $|r| < 1$ , the infinite GP converges:

S_\infty = \frac{a}{1 - r}.

|r| \ge 1

the sum does not converge (no finite value). Many problems reverse this: given

S_\infty

and one term, solve for

a

and

r

Sum to infinity

S_\infty = \frac{a}{1 - r}\quad (|r| < 1)

Worked example

Find the sum to infinity of

4 + 2 + 1 + \tfrac12 + \cdots

$a = 4,\ r = \tfrac{2}{4} = \tfrac12$ , and $|r| < 1$ so it converges.
$S_\infty = \dfrac{a}{1 - r} = \dfrac{4}{1 - \tfrac12} = \dfrac{4}{\tfrac12}$ .

Answer:

8

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the sum of

1 - \tfrac13 + \tfrac19 - \tfrac{1}{27} + \cdots

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Sum to infinity of $1 + \tfrac12 + \tfrac14 + \cdots$ ?
2.
Sum to infinity of $9 + 3 + 1 + \cdots$ ?
3.
Does $1 + 2 + 4 + \cdots$ have a finite sum?
4.
Infinite GP with $a = 5$ , $S_\infty = 10$ . Find $r$ .

From the bank · past-year question

Example 3Sequence & SeriesEASY

The sum of the series

3 - 1 + \frac{1}{3} - \frac{1}{9} + \cdots

is equal to

[Q7 · Sep · 2018]

The convergence condition is not optional

S_\infty = \tfrac{a}{1-r}

is only valid for

|r| < 1

. If a problem's ratio has

|r| \ge 1

, the sum genuinely has no finite value — there is no number to find. Always check

|r|

before reaching for the formula.

Drill 6 more on sum of an infinite gp

Concept 4 of 5

What preserves a GP

Intuition

Operations that act the same way on every term keep a GP a GP. Multiply or divide each term by a constant, raise each to the same power, or take reciprocals — all still GPs (with a new ratio). This is the GP mirror of the AP-preserving rules, and it sets up the log-bridge trick in the next subtopic.

Definition

If $a, b, c$ are in GP (so $b^2 = ac$ ), then so are:

$ka,\ kb,\ kc$ and $\tfrac{a}{k},\ \tfrac{b}{k},\ \tfrac{c}{k}$ (ratio $r$ unchanged);
$a^2,\ b^2,\ c^2$ (ratio $r^2$ ) and $\sqrt{a},\ \sqrt{b},\ \sqrt{c}$ (ratio $\sqrt{r}$ );
$\tfrac1a,\ \tfrac1b,\ \tfrac1c$ (ratio $\tfrac1r$ ).

Adding a constant to each term, however, does NOT preserve a GP.

Worked example

a, b, c

are in GP, prove that

\tfrac1a, \tfrac1b, \tfrac1c

are in GP.

GP condition on the originals: $b^2 = ac$ .
Test the three-term condition on the reciprocals: is $\left(\tfrac1b\right)^2 = \tfrac1a \cdot \tfrac1c$ ?
Left side $= \tfrac{1}{b^2}$ ; right side $= \tfrac{1}{ac}$ . Since $b^2 = ac$ , they are equal.

Answer:Yes — the reciprocals are in GP (with ratio

1/r

Practice this conceptself-check · 4 quick reps

Try it yourself

The numbers

2, 6, 18

are in GP. Are their squares

4, 36, 324

in GP?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
If $a, b, c$ are in GP, are $5a, 5b, 5c$ in GP?
2.
If $1, 2, 4$ are in GP, is $1, 4, 16$ (squares) in GP?
3.
Does adding 1 to each of $2, 4, 8$ keep a GP?
4.
Reciprocals of a GP form a?

From the bank · past-year question

Example 4Sequence & SeriesEASY

a, b, c

are in GP where

a>0, b>0, c>0

, then which of the following are correct? 1.

a^2, b^2, c^2

are in GP 2.

\frac{1}{a}, \frac{1}{b}, \frac{1}{c}

are in GP 3.

\sqrt{a}, \sqrt{b}, \sqrt{c}

are in GP Select the correct answer using the code given below:

[Q24 · Apr · 2022]

Drill 1 more on what preserves a gp

Concept 5 of 5

Product of terms and the middle-term trick

Intuition

Pair up terms of a GP from the two ends: each such pair multiplies to the same value (the square of the middle term). So the product of an odd number of terms is just the middle term raised to that count — no need to know

a

r

individually.

Definition

In a GP, terms equidistant from the ends have a constant product: $a_k \cdot a_{n+1-k} = a_1 \cdot a_n$ . Consequently the **product of the first $2m-1$ terms equals the middle term raised to the power $2m-1$ **: if the middle term is $M$ , the product is $M^{2m-1}$ . This is why "the kth term is given" is often enough to find a product.

Worked example

The 4th term of a GP is 2. Find the product of its first 7 terms.

Seven terms have a middle term — the 4th term — equal to 2.
Product of $2m - 1 = 7$ terms $= (\text{middle term})^{7} = 2^{7}$ .
$2^7 = 128$ .

Answer:

128

Practice this conceptself-check · 4 quick reps

Try it yourself

The 3rd term of a GP is 4. Find the product of its first 5 terms.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Middle term of 5 GP terms is 2; product of all five?
2.
2nd term of a 3-term GP is 5; product?
3.
Product of terms equidistant from the ends of a GP is?
4.
4th term of a 7-term GP is 3; product of all seven?

From the bank · past-year question

Example 5Sequence & SeriesEASY

The third term of a GP is 3. What is the product of its first five terms?

[Q53 · Apr · 2021]

Drill 3 more on product of terms and the middle-term trick

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

nth term, geometric mean, and the three-term condition
nth term and three-term condition
$a_n = a\,r^{\,n-1}, \qquad b^2 = ac \ \text{(for } a,b,c \text{ in GP)}$
Sum of a finite GP
Sum of n terms
$S_n = \frac{a\,(r^n - 1)}{r - 1}\quad (r \ne 1)$
Sum of an infinite GP
Sum to infinity
$S_\infty = \frac{a}{1 - r}\quad (|r| < 1)$

Watch out for (2)

Repeating-digit sums hide a GP
→ Sum of a finite GP
The convergence condition is not optional
→ Sum of an infinite GP

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Sequence & SeriesEASY

If 3rd, 8th and 13th terms of a GP are

p

q

and

r

respectively, then which one of the following is correct ?

[Q34 · Sep · 2019]

Example 2Sequence & SeriesMODERATE

A geometric progression (GP) consists of 200 terms. If the sum of odd terms of the GP is

m

, and the sum of even terms of the GP is

n

, then what is its common ratio ?

[Q8 · Sep · 2019]

Example 3Sequence & SeriesEASY

x = 1 - y + y^2 - y^3 + \ldots

where

|y| < 1

, then which is correct?

[Q23 · Apr · 2018]

Example 4Sequence & SeriesEASY

Consider the following statements: 1. If each term of a GP is multiplied by same non-zero number, then the resulting sequence is also a GP. 2. If each term of a GP is divided by same non-zero number, then the resulting sequence is also a GP. Which of the above statements is/are correct?

[Q49 · Apr · 2021]

Example 5Sequence & SeriesMODERATE

If p,

g_1

g_2

and q are in GP and m is the arithmetic mean of p and q, then

\dfrac{g_1^2}{g_2}+\dfrac{g_2^2}{g_1}

is equal to

[Q2 · Apr · 2026]

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