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Chapters
1: Goods and Services Tax (G.S.T.)
2: Banking
3: Shares and Dividends
UNIT 2: ALGEBRA
4: Linear Inequations
5: Quadratic Equation
6: Problems on Quadratic Equations
7: Ratio and Proportion
8: Remainder Theorem and Factor Theorem
9: Matrices
10: Arithmetic Progression
▶ 11: Geometric Progression
12: Reflection
13: Section and Mid-point Formulae
14: Equation of a Straight Line
UNIT 3: GEOMETRY
Chapter 15: Similarity (As a Size Transformation)
Chapter 16: Similarity of Triangles
17: Loci
Chapter 18: Angle and Cyclic Properties of a Circle
Chapter 19: Tangent Properties of Circles
20: Constructions
UNIT 4: MENSURATION
Chapter 21: Volume and Surface Area of Solids (Cylinder, Cone and Sphere)
UNIT 5: TRIGONOMETRY
Chapter 22: Trigonometrical Identities
Chapter 23: Heights and Distances
UNIT 6: STATISTICS
Chapter 24: Graphical Representation of Statistical Data
Chapter 25: Measures of Central Tendency (Mean)
Chapter 26: Median, Quartiles and Mode
UNIT 7: PROBABILITY
Chapter 27: Probability
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Solutions for Chapter 11: Geometric Progression
Below listed, you can find solutions for Chapter 11 of CISCE R.S. Aggarwal for Mathematics [English] Class 10 ICSE.
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE 11 Geometric Progression EXERCISE 11A [Pages 177 - 178]
Show that the progression 2, 6, 18, 54, 162, ......... is a G.P.
Write its (i) first term (ii) common ratio (iii) nth term (iv) 8th term.
Show that the progression $$625, 125, 25, 5, 1, \frac{1}{5}$$, ......... is a G.P.
Write its (i) first term (ii) common ratio (iii) nth term (iv) 10th term.
Show that the progression $$-27, 9, -3, 1, \frac{-1}{3}$$, ......... is a G.P.
Write its (i) first term (ii) common ratio (iii) nth term (iv) 9th term.
Show that the progression $$2, 2\sqrt{2}, 4, 4\sqrt{2}$$, ......... is a G.P.
Write its (i) first term (ii) common ratio (iii) nth term (iv) 11th term.
Show that the progression $$\frac{-3}{4}, \frac{1}{2}, \frac{-1}{3}, \frac{2}{9}$$ ......... is a G.P.
Write its (i) first term (ii) common ratio (iii) nth term (iv) 6th term.
Show that the progression 0.4, 0.8, 1.6, ......... is a G.P.
Write its (i) first term (ii) common ratio (iii) nth term (iv) 7th term.
Which term of the G.P. 3, 6, 12, 24, ..... is 768?
Which term of the G.P. 5, 10, 20, 40, ..... is 640?
Which term of the G.P. $$\sqrt{3}, 3, 3\sqrt{3}, 9$$, ..... is 729?
Find the G.P. whose 5th and 8th terms are 80 and 640 respectively.
The 4th term of a G.P. is 16 and the 7th term is 128. Find the first term and common ratio of the series
In a Geometric Progression (G.P.), the first term is 24 and the fifth term is 8. Find the ninth term of the G.P.
Find the G.P. whose 4th and 7th terms are $$\frac{1}{18}$$ and $$\frac{-1}{486}$$ respectively.
For what values of $$x$$, the numbers $$(x + 9), (x - 6)$$ and 4 are in G.P.?
Find the 6th term from the end of the G.P. $$16, 8, 4, 2, ......., \frac{1}{512}$$.
Find the 4th term from the end of the G.P. $$\frac{2}{81}, \frac{2}{27}, \frac{2}{9}, ......., 54$$.
The 4th, 6th and last term of a geometric progression are 10, 40 and 640 respectively. If the common ratio is positive, find the first term, common ratio and the number of terms of the series.
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE 11 Geometric Progression EXERCISE 11B [Pages 180 - 181]
Find the sum of first 8 terms of the G.P. 1, 3, 9, 27, 81, .........
Find the sum of first 10 terms of the G.P. $$1, \sqrt{3}, 3, 3\sqrt{3}, .......$$
Find the sum of first 9 terms of the G.P. $$1, \frac{-1}{2}, \frac{1}{4}, \frac{-1}{8}, .........$$
Find the sum of first 6 terms of the G.P. 0.1, 0.01, 0.001, .........
Find the sum of first 6 terms of the G.P. $$1, \frac{-1}{3}, \frac{1}{3^2}, \frac{-1}{3^3}, ..........$$
The first term of a G.P. is 27 and its 8th term is $$\frac{1}{81}$$. Find the sum of first seven terms of the G.P.
The 4th and the 7th terms of a G.P. are $$\frac{1}{27}$$ and $$\frac{1}{729}$$ respectively. Find the sum of first 6 terms of the G.P.
If the 6th term of a series in Geometric Progression (G.P.) is 32 and the 9th term is 256, find the:
- first term and the common ratio.
- sum of its first 10 terms.
15, 30, 60, 120.... are in G.P. (Geometric Progression):
- Find the nth term of this G.P. in terms of n.
- How many terms of the above G.P. will give the sum 945?
How many terms of the G.P. $$\frac{2}{9}, \frac{-1}{3}, \frac{1}{2}, ............$$ must be taken to make the sum equal to $$\frac{55}{72}$$?
In a G.P. the ratio of the sum of first 3 terms is to that of first 6 terms is 125 : 152. Find the common ratio.
A manufacturer reckons that the value of a machine which costs him ₹31250 depreciates each year by 20%. Find its value after 2 years.
[Hint: Value after 2 years = ₹31250 × $$\left(1 - \frac{20}{100}\right)^2$$.]
Find the sum of the following to n terms: 7 + 77 + 777 + 7777 + ..........
[Hint: $$S_n = 7(1 + 11 + 111 + 1111 +\ .........)$$ $$= \frac{7}{9} (9 + 99 + 999 + 9999 +\ .........)$$
$$= \frac{7}{9} \left[(10 - 1) + (100 - 1) + (1000 - 1)\ .........\right]$$
$$= \frac{7}{9} (10 + 100 + 1000 +\ ....... -1 -1 -1\ ........\ n \text{ times})$$]
Find the sum of n terms of a series whose mth term is $$2^m + 2m$$.
[Hint: $$S_n = \left[(2^1 + 2 \times 1) + (2^2 + 2 \times 2) + (2^3 + 2 \times 3) +\ .......\right]$$
$$= \left[(2^1 + 2^2 + 2^3 +\ ........ + 2^ \ .........\ +\ 2 \times 1 + 2 \times 2 + 2 \times 3 +\ ....)\right]$$
$$= 2^1 + 2^2 + 2^3 +\ ........ + 2(1 + 2 + 3 +\ ......)$$]
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE 11 Geometric Progression COMPETENCY-FOCUSED QUESTIONS [Pages 181 - 184]
I. Multiple Choice Questions Choose the correct alternative:
Which of the following is NOT a geometric progression?
`1/3, 1, 3, 9`
`1/5, 1/5, 1/5, 1/5`
–2, 4, –8, 16
2, 0, 4, 0, 8, 0
If the first term, common ratio and the last term of a G.P. are a, r and `l` respectively, then the nth term from the end of the G.P. is given by:
$$lr^{n - 1}$$
$$lr^{1 - n}$$
$$\frac{l}{r^{1 - n}}$$
$$\frac{l}{r^{n + 1}}$$
The sum of n terms of a G.P. with first term a and common ratio r, when r = 1, is given by:
$$S_n = \frac{a(1 - r^n)}{1 - r}$$
$$S_n = \frac{a(r^n - 1)}{r - 1}$$
$$n^2a$$
$$na$$
The general term of the G.P. $$\frac{1}{4}, -\frac{1}{2}, 1, -2, 4, ........$$ is ______.
$$(-1)^{(n - 1)} \times (2)^{(n - 3)}$$
$$(-1)^{(n - 1)} \times (2)^{(n - 2)}$$
$$(-1)^{(n - 1)} \times (-2)^{(n - 1)}$$
$$(-1)^{(n - 1)} \times (-2)^{(n - 3)}$$
The 12th term of the G.P. 2, 4, 8, 16, ........ is ______.
1024
2048
4096
8192
Which term of the G.P. $$\sqrt{3}, 3\sqrt{3}, 9\sqrt{3}, .......$$ is $$729\sqrt{3}$$?
7th
6th
9th
8th
If $$a_1, a_2, a_3, ......, a_n$$ is a G.P. having common ratio r and k is a natural number such that $$3 < k < n$$, then r is equal to ______.
$$\frac{a_k}{a_1}$$
$$\frac{a_1}{a_2}$$
$$\frac{a_k}{a_{n-3}}$$
$$\frac{a_{k-1}}{a_{k-2}}$$
The common ratio of the G.P. $$-\frac{3}{4}, \frac{1}{2}, -\frac{1}{3}, \frac{2}{9}, ......$$ is ______.
$$-\frac{4}{3}$$
$$-\frac{2}{3}$$
$$\frac{2}{3}$$
$$-\frac{3}{8}$$
The common ratio of the G.P. $$\frac{1}{a^3x^3}, ax, a^5x^5, .......$$ is ______.
$$\frac{1}{a^2x^2}$$
$$\frac{1}{a^4x^4}$$
$$a^2x^2$$
$$a^4x^4$$
The common ratio of the G.P. 0.15, 0.015, 0.00015, ...... is ______.
0.1
0.01
1
0.001
The $$n$$th term of the G.P. $$x^3, x^5, x^7, .........$$ is ______.
$$x^{(2n - 1)}$$
$$x^{(2n + 3)}$$
$$x^{(2n + 1)}$$
$$x^{3n + 2}$$
The 10th term of the G.P. $$1, -a, a^2, -a^3, ..........$$ is ______.
$$a^9$$
$$-a^{10}$$
$$-a^{11}$$
$$-a^9$$
The first term and the common ratio of the G.P. $$3, \frac{3}{2}, \frac{3}{4}, .............$$ are respectively:
3 and 2
3 and $$\frac{1}{2}$$
3 and $$\frac{3}{2}$$
$$\frac{3}{2}$$ and $$\frac{1}{2}$$
Which term of the G.P. 2, 8, 32, 128, ......... is 131072?
9th
10th
8th
12th
If the 5th term of a G.P. is 2, then the product of its first nine terms is ______.
256
1024
512
2048
If the $$(p + q)$$th and $$(p - q)$$th term of a G.P. are $$m$$ and $$n$$ respectively, then its $$p$$th term is ______.
$$mn$$
$$\sqrt{mn}$$
$$(mn)^2$$
$$\frac{m}{n}$$
If 2nd, 3rd and 6th terms of an A.P. are the three consecutive terms of a G.P., then the common ratio of the G.P. is ______.
2
3
$$\frac{1}{2}$$
$$\frac{1}{3}$$
The 6th term from the end of the G.P. 8, 4, 2, ......., $$\frac{1}{1024}$$ is ______.
$$\frac{1}{32}$$
$$\frac{1}{16}$$
$$\frac{1}{64}$$
$$\frac{1}{128}$$
The 4th term from the end of the G.P. $$\frac{2}{27}, \frac{2}{9}, \frac{2}{3}, ......., 162$$ is ______.
18
2
6
$$\frac{2}{3}$$
The product of first three terms of a G.P. is $$-1$$ and the common ratio is $$-\frac{3}{4}$$. The sum of these three terms is ______.
$$\frac{12}{13}$$
$$\frac{11}{13}$$
$$\frac{11}{12}$$
$$\frac{13}{12}$$
For what values of $$x$$ are the numbers $$-\frac{2}{7}, x, -\frac{7}{2}$$ in G.P.?
0, 1
0, –1
–1, 1
–2, 2
How many terms of the G.P. 1, 4, 16, 64, ........ will make the sum 5461?
6
9
7
8
The sum of 7 terms of the G.P. 3, 6, 12, ........... is ______.
181
241
381
421
The sum of the first two terms of a G.P. is $$-4$$ and the fifth term is 4 times the third term. Then, the first term of the G.P. is ______.
$$-\frac{3}{4}$$ or 4
$$\frac{3}{4}$$ or $$\frac{1}{4}$$
$$\frac{4}{3}$$ or $$\frac{1}{4}$$
$$-\frac{4}{3}$$ or 4
If x, y and z are in G.P., then the relation between x, y and z can be:
y = x + z
y = xz
2y = x + z
$$y = \sqrt{xz}$$
The product of first five terms of a G.P. with first term a and common ratio r > 1 is equal to ______.
$$\frac{a(r^{5} - 1)}{(r - 1)}$$
$$ar^{4}$$
$$a^{5}r^{10}$$
$$\frac{a(r^{5} - 1)}{r}$$
If 5th, 8th and 11th terms of a G.P. are x, y and z respectively, then, which one of the following is correct?
$$y^{2} = x^{2}z^{2}$$
$$y^{2} = x^{2} + z^{2}$$
$$y^{2} = xz$$
$$xyz = 1$$
Consider the G.P. $$a, ar, ar^{2}, ........, l$$. The kth term from the end is ______.
$$\frac{l}{r^{k - 1}}$$
$$\frac{l}{k}$$
$$\frac{al}{r^{k}}$$
$$\frac{l}{ar^{k - 1}}$$
II. Assertion-Reason Type Questions Directions: In each question below, there are two statements labelled as Assertion (A) and Reason (R). Choose the correct answer as per the options given below:
Assertion (A): The nth term of a G.P. is given by $$T_{n} = ar^{n - 1}$$.
Reason (R): A sequence $$a_{1}, a_{2}, a_{3} ....$$ is said to a G.P. if $$\frac{a_{1}}{a_{2}} = \frac{a_{2}}{a_{3}} = \frac{a_{3}}{a_{4}} ....$$ = constant known as the common ratio.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.
Assertion (A): The 50th term from the end of the G.P. $$4, 6, 9, \frac{27}{2}, .... \frac{6561}{64}$$ is $$\frac{27}{2}$$.
Reason (R): nth term from the end of a G.P. is given by $$\frac{l}{r^{n - 1}}$$.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.
Assertion (A): The sum of first 10 terms of the G.P. 3, 6, 9, 12, ... is 3096.
Reason (R): The sum of first n-terms of a G.P. is given by $$S_{n} = \frac{a(r^{n} - 1)}{r - 1}$$.
Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true, but R is false.
A is false, but R is true.
III. Analytical and Application Based Questions
Write the first five terms of the sequence given by `(sqrt(3))^n, n ∈ N`.
- Is the sequence an A.P. or G.P?
- If the sum of its first ten terms is `p(3 + sqrt(3))`, find the value of p.
IV. Case Study Based Questions Read the following case studies carefully and answer the questions based on them:
| The number of bacteria in a certain culture doubles every hour. There were 30 bacteria present in the culture originally. |
1. The number of bacteria that would be present at the end of 4th hour is:
(a) 240
(b) 480
(c) 960
(d) 450
2. The number of bacteria that would be present at the end of 10th hour is:
(a) 30720
(b) 15360
(c) 61440
d) 27540
3. The number of bacteria that would be present at the end of nth hour is:
(a) `30 xx 2^((n - 1))`
(b) `30 xx 2^((n + 1))`
(c) `30 xx 2^n`
(d) none of these
4. What would be the sum of the first 5 terms of the G.P. that is formed by the number of bacteria after the completion of each hour?
(a) 450
(b) 930
(c) 1890
(d) none of these
5. What would be the sum of the first n terms of the G.P. that is formed by the number of bacteria after the completion of each hour?
(a) `30 xx 2^n - 1`
(b) `30 xx (2^(n - 1) - 1)`
(c) `30 xx (2^(n + 1) - 1)`
(d) `30 xx 2^n - 30`
| A man writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with the instruction that they move the chain similarly. Assume that the chain is not broken and it costs ₹4 to mail one letter. |
1. The number of letters mailed in the 6th set of letters is:
(a) 2048
(b) 8192
(c) 4096
(d) 1024
2. The amount spent on the postage of 6th set of letters is:
(a) ₹16384
(b) ₹8192
(c) ₹32768
(d) ₹4096
3. The total number of letters mailed till the 5th set of letters is:
(a) 1024
(b) 4092
(c) 1236
(d) 1364
4. The amount spent on the postage till the 5th set of letters is:
(a) ₹16368
(b) ₹ 4096
(c) ₹5456
(d) ₹ 4944
5. The amount spent on the postage till the nth set of letters is:
(a) ₹ `[4/3 (4^n - 4)]`
(b) ₹ `[16/3 (4^n - 1)]`
(a) ₹ `[2/3 (4^n - 1)]`
(a) ₹ `[16 (4^n - 1)]`
Solutions for 11: Geometric Progression
R.S. Aggarwal solutions for Mathematics [English] Class 10 ICSE chapter 11 - Geometric Progression
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