1. Algebra — Equations, Progressions and Series

Geometric progression

Q: FAQ 1: Can a geometric progression have negative or zero terms?

Terms can be negative if the common ratio $r$ is negative (resulting in alternating signs). However, the first term $a$ and common ratio $r$ cannot be zero, as division by zero is undefined and zero terms eliminate the multiplicative structure.

Q: FAQ 2: What is the most common reason students lose marks on G.P. questions?

Candidates frequently fail to check the convergence condition $|r| < 1$ before applying the infinite sum formula $S_\infty = \frac{a}{1-r}$, leading to nonsensical results when applied to divergent series.

Q: FAQ 3: How can candidates solve G.P. product and sum problems faster during the exam?

Always assume symmetric terms ($\frac{a}{r}, a, ar$ for 3 terms) when the product is given, and utilize logarithmic transformations or index properties to bypass lengthy polynomial expansions.

A geometric progression (G.P.) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

General Term and Sum of a Geometric Progression

Let a sequence be given by $a_1, a_2, a_3, \dots, a_n, \dots$ such that $\frac{a_{n+1}}{a_n} = r$ (constant) for all $n \ge 1$, where $a_1 = a \neq 0$ and $r \neq 0, 1$. Then the $n$-th term is given by $a_n = a r^{n-1}$, and the sum of the first $n$ terms is $S_n = \frac{a(r^n - 1)}{r - 1}$ for $r > 1$ and $S_n = \frac{a(1 - r^n)}{1 - r}$ for $r < 1$.

📐 Geometric Interpretation

Points plotted as $(n, a_n)$ on a logarithmic scale form a straight line, while on a Cartesian plane they lie on an exponential curve $y = a \cdot r^{x-1}$, exhibiting rapid divergence for $|r| > 1$ and asymptotic convergence to zero for $|r| < 1$.

⚙️ Algebraic & Structural Meaning

Represents the discrete analogue of the exponential function $f(x) = a e^{kx}$, satisfying the first-order recurrence relation $a_{n+1} - r a_n = 0$, forming a cyclic field structure under multiplication.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Non-zero First Term	`Discrete set $a \in \mathbb{R} \setminus \{0\}$`	Strictly Mandatory / Necessary	If $a = 0$, all terms vanish, rendering the sequence trivial and the common ratio undefined.
Constant Common Ratio	`Ratio $r \in \mathbb{R} \setminus \{0\}$`	Strictly Mandatory	If $r$ varies with $n$, the progression ceases to be geometric, transitioning into an arithmetico-geometric or general recursive sequence.
Exclusion of Unity for Ratio in Infinite Sums	`$\|r\| < 1$ for convergence`	Essential for conclusion	If $\|r\| \ge 1$, the infinite series $\sum_{n=1}^\infty a r^{n-1}$ diverges or oscillates finitely/infinitely, making the sum to infinity $S_\infty$ undefined.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Symmetric Terms Selection

$$ \text{For 3 terms: } \frac{a}{r}, a, ar \quad | \quad \text{For 4 terms: } \frac{a}{r^3}, \frac{a}{r}, ar, ar^3 $$

Rule: When the product of terms is given, always assume symmetric terms about the middle term to eliminate variables instantly during multiplication.

Application: Use this to find numbers whose product and sum are given.

Infinite G.P. Sum Formula

$$ S_\infty = \frac{a}{1 - r}, \quad \text{where } |r| < 1 $$

Rule: Use this direct rationalization formula for recurring decimals or infinite geometric series without computing partial sums.

Application: Standard recurring decimal conversion and infinite series sum questions in DSSSB and KVS.

Property of Equidistant Terms

$$ a_k \cdot a_{n-k+1} = a_1 \cdot a_n = a_2 \cdot a_{n-1} = \text{constant} $$

Rule: The product of terms equidistant from the beginning and the end of a finite G.P. is constant and equal to the product of the first and last terms.

Logarithmic Transformation

$$ \text{If } a, b, c \in \text{G.P.} \implies \log a, \log b, \log c \in \text{A.P.} $$

Rule: Apply logarithms to convert multiplicative G.P. problems into linear A.P. problems.

Sum of Powers Property

$$ \text{If } a_1, a_2, \dots, a_n \in \text{G.P.}, \text{ then } a_1^k, a_2^k, \dots, a_n^k \text{ also form a G.P. with ratio } r^k $$

Rule: Raise each term to power $k$ and update the common ratio directly to $r^k$ instead of recalculating.

Rapid Option Elimination via Ratio Test

$$ \frac{T_{n+1}}{T_n} = \text{Constant } r $$

Rule: Test the ratio of the 2nd term to the 1st term in the given options; eliminate options where this ratio is variable.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

The Common Ratio Sign Ambiguity

Common Pitfall: Assuming the common ratio $r$ must always be positive. Candidates forget that alternating signs indicate a negative common ratio.

Counterexample: Sequence: $2, -4, 8, -16, \dots \implies r = -2

Why it fails: Treating $r$ as positive leads to sign errors in finding $n$-th terms or sums involving even/odd powers.

🛡️ Exam Defense: Always compute $r$ as $T_2 / T_1$ with exact signs, and check alternating behavior.

⚠️ COMMON ERROR #02

Divergent Infinite Series Sum Formula Misapplication

Common Pitfall: Applying $S_\infty = \frac{a}{1-r}$ when $|r| \ge 1$.

Counterexample: \text{For } 1 + 2 + 4 + 8 + \dots, \text{ here } r = 2 \ge 1.

Why it fails: Formula yields a negative or fractional value (e.g., $1/(1-2) = -1$), which is absurd for a strictly positive diverging series.

🛡️ Exam Defense: Mandatory check: Always verify $|r| < 1$ before deploying the infinite sum formula.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Arithmetic Progression (A.P.)	Constant difference $d = a_{n+1} - a_n$	$T_n = a + (n-1)d$	Points lie on a straight line $y = mx + c$	Keywords: 'sum of terms', 'constant difference'
Geometric Progression (G.P.)	Constant ratio $r = a_{n+1}/a_n$	$T_n = a r^{n-1}$	Points lie on an exponential curve $y = a r^x$	Keywords: 'product of terms', 'constant ratio'
Harmonical Progression (H.P.)	Reciprocals form an A.P.	$T_n = \frac{1}{a + (n-1)d}$	Points form harmonic coordinates	Keywords: 'reciprocals', 'harmonic mean'
Arithmetico-Geometric Progression (AGP)	Product of A.P. and G.P. terms	$T_n = [a + (n-1)d]r^{n-1}$	Combined linear-exponential growth	Keywords: 'series with products of AP and GP terms'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If the 4th, 7th, and 10th terms of a G.P. are $p, q,$ and $s$ respectively, then prove that $q^2 = ps$.

(A) $q = ps$(B) $q^2 = ps$(C) $q = p^2s$(D) $pq = s^2$

📚 Method A: Formal Textbook Derivation

Let the first term be $a$ and common ratio be $r$. Then $p = a r^3$, $q = a r^6$, and $s = a r^9$. Consider $ps = (a r^3)(a r^9) = a^2 r^{12} = (a r^6)^2 = q^2$. Hence $q^2 = ps$.

⚡ Method B: 25-Second Exam Shortcut

Notice the indices 4, 7, 10 form an A.P. ($4 + 10 = 2 \times 7$). Therefore, the corresponding terms in G.P. satisfy $p \cdot s = q^2$ instantly.

💡 Tip: Whenever indices of terms in a G.P. form an A.P., their corresponding terms form a property related to geometric mean.

CANONICAL PROBLEM 02 Dual-Method Breakdown

Find the sum to infinity of the series $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots$.

(A) $\frac{3}{2}$(B) $2$(C) $\frac{1}{2}$(D) $3$

📚 Method A: Formal Textbook Derivation

Here $a = 1$ and $r = \frac{1}{3}$. Since $|r| = \frac{1}{3} < 1$, apply $S_\infty = \frac{a}{1 - r} = \frac{1}{1 - 1/3} = \frac{1}{2/3} = \frac{3}{2}$.

⚡ Method B: 25-Second Exam Shortcut

Direct substitution into $S_\infty = \frac{a}{1-r}$ yields $\frac{1}{1 - 1/3} = \frac{3}{2}$.

💡 Tip: Check first term and common ratio carefully; watch out for alternating signs.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If the sum of first 3 terms of a G.P. is $\13/12$ and their product is $-1$, then the common ratio $r$ is:

A $\frac{4}{3}$ or $\frac{3}{4}$

B $-\frac{4}{3}$ or $-\frac{3}{4}$

C $\frac{1}{2}$ or $2$

D $-\frac{1}{2}$ or $-2$

Derivation & Analysis:

Let terms be $\frac{a}{r}, a, ar$. Product $= a^3 = -1 \implies a = -1$. Sum $= -\frac{1}{r} - 1 - r = \frac{13}{12} \implies -(\frac{1}{r} + r) = \frac{25}{12} \implies r = -\frac{4}{3}$ or $-\frac{3}{4}$.

MCQ 2: If $a, b, c$ are in G.P., then $\frac{a^2 + ab + b^2}{b^2 + bc + c^2}$ is equal to:

A $\frac{a}{c}$

B $\frac{b}{c}$

C $\frac{a}{b}$

D $\frac{c}{a}$

Derivation & Analysis:

Since $a, b, c$ are in G.P., $b = ar$ and $c = ar^2$. Substitute into expression: $\frac{a^2 + a(ar) + (ar)^2}{(ar)^2 + (ar)(ar^2) + (ar^2)^2} = \frac{a^2(1 + r + r^2)}{a^2 r^2(1 + r + r^2)} = \frac{1}{r^2} = \frac{a^2}{b^2} = \frac{a}{b}$ (using $b^2 = ac$).

MCQ 3: The sum of the infinite series $5 + \frac{5}{5} + \frac{5}{25} + \frac{5}{125} + \dots$ is:

A $\frac{25}{4}$

B $\frac{5}{4}$

C $\frac{4}{25}$

D $6$

Derivation & Analysis:

Series is $5 + 1 + \frac{1}{5} + \frac{1}{25} + \dots$. Here $a = 5, r = \frac{1}{5}$. $S_\infty = \frac{5}{1 - 1/5} = \frac{5}{4/5} = \frac{25}{4}.

MCQ 4: If $x, y, z$ are in A.P. and $3^x, 3^y, 3^z$ are in G.P., then:

A $x = y = z$

B $x, y, z$ are all zero

C $2y = x + z$ and $3^y = \sqrt{3^x 3^z}$

D $x = y$ only

Derivation & Analysis:

Given $2y = x + z$ (A.P.) and $(3^y)^2 = 3^x \cdot 3^z \implies 2y = x + z$ (which is identically satisfied). Also exponential property holds for any A.P. Thus $x, y, z$ can be any A.P., but if $3^x, 3^y, 3^z$ form a G.P., $x, y, z$ must be equal for specific integer constraints or general equality holds if $d=0$. More precisely, $x, y, z$ form an A.P. automatically. If $3^x, 3^y, 3^z$ are G.P., $2y = x+z$ and $3^x, 3^y, 3^z$ implies $x=y=z$ for distinct real roots unless $d=0$.

MCQ 5: Which of the following sequences is a geometric progression?

A $2, 4, 6, 8, \dots$

B $3, 9, 27, 81, \dots$

C $1, 4, 9, 16, \dots$

D $1, 2, 4, 7, \dots$

Derivation & Analysis:

Checking ratios: (B) $\frac{9}{3} = 3, \frac{27}{9} = 3, \frac{81}{27} = 3$. Constant ratio $r = 3$.

MCQ 6: If the $n$-th term of a G.P. is $2^{n-1}$, then the sum of its first 5 terms is:

A $31$

B $32$

C $15$

D $63$

Derivation & Analysis:

Here $a_n = 1 \cdot 2^{n-1} \implies a = 1, r = 2$. Sum of first 5 terms $S_5 = \frac{1(2^5 - 1)}{2 - 1} = 32 - 1 = 31$.

MCQ 7: The geometric mean of two numbers $a$ and $b$ is 4, and their arithmetic mean is $\frac{17}{4}$. The numbers are:

A $1, 16$

B $2, 8$

C $4, 4$

D $16, 1$

Derivation & Analysis:

$\sqrt{ab} = 4 \implies ab = 16$. $\frac{a+b}{2} = \frac{17}{4} \implies a+b = \frac{17}{2}$. Solving $a=1, b=16$ or $16, 1$.

MCQ 8: Assertion (A): The sum of an infinite geometric series with first term $2$ and common ratio $\frac{1}{2}$ is $4$. Reason (R): $S_\infty = \frac{a}{1-r}$ is valid for all real $r$.

A Both (A) and (R) are true and (R) is the correct explanation of (A)

B Both (A) and (R) are true but (R) is not the correct explanation of (A)

C (A) is true but (R) is false

D (A) is false but (R) is true

Derivation & Analysis:

Assertion is true ($S_\infty = \frac{2}{1 - 1/2} = 4$). Reason is false because $S_\infty formula$ is only valid when $|r| < 1$, not for all real $r$.

MCQ 9: If $x = 1 + a + a^2 + \dots$ to $\infty$ and $y = 1 + b + b^2 + \dots$ to $\infty$, where $|a| < 1$ and $|b| < 1$, then $1 + ab + a^2b^2 + \dots$ to $\infty$ is equal to:

A $\frac{xy}{x+y-1}$

B $\frac{x+y}{xy}$

C $\frac{xy}{x+y}$

D $\frac{x-y}{xy}$

Derivation & Analysis:

$x = \frac{1}{1-a} \implies 1-a = \frac{1}{x} \implies a = \frac{x-1}{x}$. Similarly $b = \frac{y-1}{y}$. Series sum $S = \frac{1}{1-ab}$. Substitute $ab$ and simplify to get $\frac{xy}{x+y-1}$.

Q10

MCQ 10: If $p, q, r$ are in G.P., then equations $px^2 + 2qx + r = 0$ and $dx^2 + 2ex + f = 0$ have a common root if $\frac{d}{p}, \frac{e}{q}, \frac{f}{r}$ are in:

A A.P.

B G.P.

C H.P.

D None of these

Derivation & Analysis:

Since $p, q, r$ are in G.P., $q^2 = pr$. The roots of $px^2 + 2qx + r = 0$ are equal because discriminant $4q^2 - 4pr = 0$. For the two equations to share a root, the root must satisfy both. Testing coefficient relations yields $\frac{d}{p}, \frac{e}{q}, \frac{f}{r}$ in A.P.

Common questions

Terms can be negative if the common ratio $r$ is negative (resulting in alternating signs). However, the first term $a$ and common ratio $r$ cannot be zero, as division by zero is undefined and zero terms eliminate the multiplicative structure.

Candidates frequently fail to check the convergence condition $|r| < 1$ before applying the infinite sum formula $S_\infty = \frac{a}{1-r}$, leading to nonsensical results when applied to divergent series.

Always assume symmetric terms ($\frac{a}{r}, a, ar$ for 3 terms) when the product is given, and utilize logarithmic transformations or index properties to bypass lengthy polynomial expansions.