LFM Pure and Mechanics

gaokao 2019 Q18 12 marks Arithmetic-Geometric Hybrid Problem View

18. (12 points)
Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with all positive terms, $a _ { 1 } = 2 , a _ { 3 } = 2 a _ { 2 } + 16$.
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $b _ { n } = \log _ { 2 } a _ { n }$, find the sum of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$.

gaokao 2020 Q10 5 marks Finite Geometric Sum and Term Relationships View

Let $\left\{ a _ { n } \right\}$ be a geometric sequence with $a _ { 1 } + a _ { 2 } + a _ { 3 } = 1 , a _ { 2 } + a _ { 3 } + a _ { 4 } = 2$ , then $a _ { 6 } + a _ { 7 } + a _ { 8 } =$
A. 12
B. 24
C. 30
D. 32

gaokao 2020 Q17 12 marks Derive General Term from Geometric Property View

Let the geometric sequence $\left\{ a _ { n } \right\}$ satisfy $a _ { 1 } + a _ { 2 } = 4 , a _ { 3 } - a _ { 1 } = 8$ .
(1) Find the general term formula for $\left\{ a _ { n } \right\}$;
(2) Let $S _ { n }$ be the sum of the first $n$ terms of the sequence $\left\{ \log _ { 3 } a _ { n } \right\}$. If $S _ { m } + S _ { m + 1 } = S _ { m + 3 }$, find $m$ .

gaokao 2021 Q7 Logical Relationship Between Conditions on Geometric Sequences View

7. For a geometric sequence $\{a_n\}$ with common ratio $q$ and sum of the first $n$ terms $S_n$, let Proposition A: $q > 0$. Proposition B: $\{S_n\}$ is an increasing sequence. Then
A. A is a sufficient but not necessary condition for B
B. A is a necessary but not sufficient condition for B
C. A is a necessary and sufficient condition for B
D. A is neither a sufficient nor a necessary condition for B

gaokao 2021 Q9 Finite Geometric Sum and Term Relationships View

9. Let $S _ { n }$ denote the sum of the first $n$ terms of the geometric sequence $\left\{ a _ { n } \right\}$. If $S _ { 2 } = 4 , S _ { 4 } = 6$, then $S _ { 6 } =$
A. 7
B. 8
C. 9
D. 10

gaokao 2022 Q8 5 marks Finite Geometric Sum and Term Relationships View

Given that the sum of the first 3 terms of a geometric sequence $\{a_n\}$ is $168$, and $a_2 - a_5 = 42$, then $a_6 =$
A. $14$
B. $12$
C. $6$
D. $3$

gaokao 2022 Q10 5 marks Finite Geometric Sum and Term Relationships View

Given that the geometric sequence $\left\{ a _ { n } \right\}$ has the sum of its first 3 terms equal to 168 , and $a _ { 2 } - a _ { 5 } = 42$ , then $a _ { 6 } =$
A. 14
B. 12
C. 6
D. 3

gaokao 2023 Q8 5 marks Finite Geometric Sum and Term Relationships View

Let $S_n$ denote the sum of the first $n$ terms of the geometric sequence $\{a_n\}$. If $S_4=-5$, $S_6=21S_2$, then $S_8=$
A. 120
B. 85
C. $-85$
D. $-120$

gaokao 2023 Q15 Finite Geometric Sum and Term Relationships View

Given that $\left\{ a _ { n } \right\}$ is a geometric sequence with $a _ { 2 } a _ { 4 } a _ { 5 } = a _ { 3 } a _ { 6 }$ and $a _ { 9 } a _ { 10 } = - 8$, then $a _ { 7 } = $ \_\_\_\_

gaokao 2024 Q14 5 marks Finite Geometric Sum and Term Relationships View

Given that the volumes of three cylinders form a geometric sequence with common ratio 10. The diameter of the first cylinder is 65 mm, the diameters of the second and third cylinders are 325 mm, and the height of the third cylinder is 230 mm. Find the heights of the first two cylinders respectively as \_\_\_\_.

gaokao 2025 Q9 6 marks True/False or Multiple-Statement Verification View

Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, and let $q$ be the common ratio of $\{a_n\}$, $q > 0$. If $S_3 = 7$, $a_3 = 1$, then
A. $q = \frac{1}{2}$
B. $a_5 = \frac{1}{9}$
C. $S_5 = 8$
D. $a_n + S_n = 8$

gaokao 2025 Q13 5 marks Finite Geometric Sum and Term Relationships View

If a positive geometric sequence has the sum of its first 4 terms equal to $4$ and the sum of its first 8 terms equal to $68$, then the common ratio of the geometric sequence is $\_\_\_\_$ .

gaokao 2025 Q13 5 marks Finite Geometric Sum and Term Relationships View

If the sum of the first 4 terms of a geometric sequence is 4 and the sum of the first 8 terms is 68, then the common ratio of the geometric sequence is $\_\_\_\_$ .

grandes-ecoles 2021 Q1 Proof of a Structural Property of Geometric Sequences View

Show that a geometric sequence is hypergeometric.

iran-konkur 2022 Q101 Finite Geometric Sum and Term Relationships View

101-- Geometric sequences with a common ratio greater than one that include 5 terms and are members of the set $\{1, 2, \ldots, 100\}$. How many of these sequences can be found whose terms are all members of the set $\{1, 2, \ldots, 100\}$?
(1) $2$ (2) $4$ (3) $6$ (4) $7$

isi-entrance None Q8 Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

Let $\{C_n\}$ be an infinite sequence of circles lying in the positive quadrant of the $XY$-plane, with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches both $X$-axis and the $Y$-axis. Further, for all $n \geq 1$, the circle $C_{n+1}$ touches the circle $C_n$ externally. If $C_1$ has radius 10 cm, then show that the sum of the areas of all these circles is $\frac{25\pi}{3\sqrt{2}-4}$ sq. cm.

isi-entrance 2013 Q44 4 marks Arithmetic-Geometric Sequence Interplay View

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above.

isi-entrance 2016 Q44 4 marks Arithmetic-Geometric Sequence Interplay View

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then $\frac{d}{a}, \frac{e}{b}$ and $\frac{f}{c}$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above

isi-entrance 2016 Q44 4 marks Arithmetic-Geometric Sequence Interplay View

Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in
(A) A.P.
(B) G.P.
(C) H.P.
(D) none of the above

isi-entrance 2020 Q18 Sum of an Infinite Geometric Series (Direct Computation) View

Which of the following is the sum of an infinite geometric sequence whose terms come from the set $\left\{ 1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \frac { 1 } { 2 ^ { n } } , \ldots \right\}$ ?
(A) $\frac { 1 } { 5 }$
(B) $\frac { 1 } { 7 }$
(C) $\frac { 1 } { 9 }$
(D) $\frac { 1 } { 11 }$

isi-entrance 2026 Q12 Arithmetic-Geometric Sequence Interplay View

Suppose $a , b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in
(a) A.P.
(B) G.P.
(C) H.P.
(D) none of the above.

jee-advanced 1999 Q33 Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

33. Let $S 1 , S 2 \ldots$ be squares such that for each $n ^ { 3 } 1$, the length of a side of Snequals the length of a diagonal of $\mathrm { Sn } + 1$. If the length of a side of S 1 is 10 cm , then for which of the following values of $n$ is the area of $S n$ less than $1 \mathrm { sq } . \mathrm { cm }$ ?
(A) 7
(B) 8
(C) 9
(D) 10

jee-advanced 2000 Q12 Determine the Limit of a Sequence via Geometric Series View

12. For $x \in R$, $\operatorname { limn } \rightarrow \infty ( ( x - 3 ) / ( x + 2 ) ) x =$
(A) e
(B) $e - 1$
(C) $e - 5$
(D) e 5

jee-advanced 2001 Q15 Arithmetic-Geometric Sequence Interplay View

15. Let $a$ and $\beta$ be the roots of $x 2 - x + p = 0$ and $y$ and $\delta$ be the roots of $x 2 - 4 x + q = 0$. if $\mathrm { a } , \beta , \mathrm { Y } , \delta$ are in G.P. then the integral values of P and q respectively, are:
(A) $- 2 , - 32$
(B) $- 2,3$
(C) $- 6,3$
(D) $- 6 , - 32$

jee-advanced 2001 Q33 Addition/Subtraction Formula Evaluation View

33. If $a + \beta = \pi / 2$ and $\beta + \gamma = a$, then tan $a$ equals:
(A) $2 ( \tan \beta + \tan \gamma )$
(B) $\tan \beta + \tan \gamma$
(C) $\tan \beta + 2 \tan \gamma$
(D) $2 \tan \beta + \tan \gamma$

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LFM Pure and Mechanics

Indices and Surds 96

Exponential Functions 197

Laws of Logarithms 242

Exponential Equations & Modelling 28

Arithmetic Sequences and Series 363

Geometric Sequences and Series 159

Differentiation from First Principles 45

Tangents, normals and gradients 140

Stationary points and optimisation 505

Applied differentiation 76

Indefinite & Definite Integrals 502

Areas Between Curves 71

Areas by integration 221

Volumes of Revolution 68

Numerical integration 49

Vectors Introduction & 2D 211

Vectors 3D & Lines 524

Travel graphs 14

SUVAT in 2D & Gravity 31

Constant acceleration (SUVAT) 78

Variable acceleration (1D) 55

Variable acceleration (vectors) 32

Forces, equilibrium and resultants 24

Newton's laws and connected particles 21

Pulley systems 6

Motion on a slope 6

Friction 8

Momentum and Collisions 34

Moments 28

Parametric curves and Cartesian conversion 9

Parametric differentiation 13

Parametric integration 6

Projectiles 49