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Mathematics · Class 11 Science

Ch 11Sequences and Series — Class 11 Mathematics, concept-first.

A sequence is a set of numbers where the numbers are arranged in a definite order, like the natural numbers, the even integers between 10 and 100, or the squares of integers. In general, a sequence is written as where is the first term, is the fourth term, and so on up to , the th term.

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2.1

Sequence

A sequence is a set of numbers where the numbers are arranged in a definite order, like the natural numbers, the even integers between 10 and 100, or the squares of integers.

+EXERCISE 2.122 questions
  1. Q1Check whether the following sequence is a G.P. If so, write $t_n$: $2, 6, 18, 54, \ldots$Free
  2. Q2Check whether the following sequence is a G.P. If so, write $t_n$: $1, -5, 25, -125, \ldots$Free
  3. Q3Check whether the following sequence is a G.P. If so, write $t_n$: $\sqrt5, \dfrac{1}{\sqrt5}, \dfrac{1}{5\sqrt5}, \dfrac{1}{25\sqrt5}, \ldo…Free
  4. Q4Check whether the following sequence is a G.P. If so, write $t_n$: $3, 4, 5, 6, \ldots$Preview
  5. Q5Check whether the following sequence is a G.P. If so, write $t_n$: $7, 14, 21, 28, \ldots$Preview
  6. Q6For the G.P., if $r=\dfrac13$, $a=9$, find $t_7$.Preview
  7. Q7For the G.P., if $a=\dfrac{7}{243}$, $r=3$, find $t_6$.Preview
  8. Q8For the G.P., if $r=-3$ and $t_6=1701$, find $a$.Preview
  9. Q9For the G.P., if $a=\dfrac23$, $t_6=162$, find $r$.Preview
  10. Q10Which term of the G.P. $5, 25, 125, 625, \ldots$ is $5^{10}$?Preview
  11. Q11For what values of $x$, the terms $\dfrac43, x, \dfrac{4}{27}$ are in G.P.?Preview
  12. Q12If for a sequence, $t_n=\dfrac{5^{n-3}}{2^{n-3}}$, show that the sequence is a G.P. Find its first term and the common ratio.Preview
  13. Q13Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.Preview
  14. Q14Find four numbers in G.P. such that sum of the middle two numbers is $\dfrac{10}{3}$ and their product is 1.Preview
  15. Q15Find five numbers in G.P. such that their product is 1024 and fifth term is square of the third term.Preview
  16. Q16The fifth term of a G.P. is $x$, eighth term of a G.P. is $y$ and eleventh term of a G.P. is $z$. Verify whether $y^2=xz$.Preview
  17. Q17If $p, q, r, s$ are in G.P. show that $p+q, q+r, r+s$ are also in G.P.Preview
  18. Q18The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there…Preview
  19. Q19A ball is dropped from a height of 80 ft. The ball is such that it rebounds $\dfrac34$th of the height it has fallen. How high does the ball…Preview
  20. Q20The numbers $3, x$ and $x+6$ are in G.P. Find (i) $x$, (ii) 20th term, (iii) $n$th term.Preview
  21. Q21Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning, write down the number of mosquitoes after (i)…Preview
  22. Q22The numbers $x-6, 2x$ and $x^2$ are in G.P. Find (i) $x$ (ii) 1st term (iii) $n$th term.Preview
2.2

Arithmetic Progression (A.P.)

In a sequence, if the difference between any term and its preceding term () is constant, the sequence is called an Arithmetic Progression (A.P.).

+EXERCISE 2.223 questions
  1. Q23For the following G.P., find $S_n$: $3, 6, 12, 24, \ldots$Free
  2. Q24For the following G.P., find $S_n$: $p, q, \dfrac{q^2}{p}, \dfrac{q^3}{p^2}, \ldots$Free
  3. Q25For the following G.P., find $S_n$: $0.7, 0.07, 0.007, \ldots$Free
  4. Q26For the following G.P., find $S_n$: $\sqrt5, -5, 5\sqrt5, -25, \ldots$Preview
  5. Q27For a G.P., $a=2$, $r=-\dfrac23$, find $S_6$.Preview
  6. Q28For a G.P., if $S_5=1023$, $r=4$, find $a$.Preview
  7. Q29For a G.P., if $a=2$, $r=3$, $S_n=242$, find $n$.Preview
  8. Q30For a G.P., sum of first 3 terms is 125 and sum of next 3 terms is 27, find the value of $r$.Preview
  9. Q31For a G.P., if $t_3=20$, $t_6=160$, find $S_7$.Preview
  10. Q32For a G.P., if $t_4=16$, $t_9=512$, find $S_{10}$.Preview
  11. Q33Find the sum to $n$ terms: $3+33+333+3333+\ldots$Preview
  12. Q34Find the sum to $n$ terms: $8+88+888+8888+\ldots$Preview
  13. Q35Find the sum to $n$ terms: $0.4+0.44+0.444+\ldots$Preview
  14. Q36Find the sum to $n$ terms: $0.7+0.77+0.777+\ldots$Preview
  15. Q37Find the sum to $n$ terms of the sequence $0.5, 0.05, 0.005, \ldots$Preview
  16. Q38Find the sum to $n$ terms of the sequence $0.2, 0.02, 0.002, \ldots$Preview
  17. Q39For a sequence, if $S_n=2(3^n-1)$, find the $n$th term, hence show that the sequence is a G.P.Preview
  18. Q40If $S, P, R$ are the sum, product and sum of the reciprocals of $n$ terms of a G.P. respectively, then verify that $\left(\dfrac{S}{R}\right…Preview
  19. Q41If $S_n, S_{2n}, S_{3n}$ are the sum of $n, 2n, 3n$ terms of a G.P. respectively, then verify that $S_n(S_{3n}-S_{2n})=(S_{2n}-S_n)^2$.Preview
  20. Q42Find $\displaystyle\sum_{r=1}^{10}(3^r-2^r)$.Preview
  21. Q43Find $\displaystyle\sum_{r=1}^{10}(5^r-3^r)$.Preview
  22. Q44The value of a house appreciates 5% per year. How much is the house worth after 6 years if its current worth is Rs. 15 Lac. [Given: $(1.05)^…Preview
  23. Q45If one invests Rs. 10,000 in a bank at a rate of interest 8% per annum, how long does it take to double the money by compound interest? [$(1…Preview
2.3

Geometric Progression

A sequence is a Geometric Progression (G.P.) if the ratio of any term to the one before it, , is constant for all ; is called the common ratio. A G.P.

+EXERCISE 2.318 questions
  1. Q46Determine whether the sum to infinity of the following G.P. exists; if it exists find it: $\dfrac12, \dfrac14, \dfrac18, \dfrac1{16}, \ldots…Free
  2. Q47Determine whether the sum to infinity of the following G.P. exists; if it exists find it: $2, \dfrac43, \dfrac89, \dfrac{16}{27}, \ldots$Free
  3. Q48Determine whether the sum to infinity of the following G.P. exists; if it exists find it: $-3, 1, -\dfrac13, \dfrac19, \ldots$Free
  4. Q49Determine whether the sum to infinity of the following G.P. exists; if it exists find it: $\dfrac15, -\dfrac25, \dfrac45, -\dfrac85, \dfrac{…Preview
  5. Q50Determine whether the sum to infinity of the following G.P. exists; if it exists find it: $9, 8.1, 7.29, \ldots$Preview
  6. Q51Express the following recurring decimal as a rational number: $0.\overline{7}$Preview
  7. Q52Express the following recurring decimal as a rational number: $2.\overline{4}$Preview
  8. Q53Express the following recurring decimal as a rational number: $2.\overline{35}$Preview
  9. Q54Express the following recurring decimal as a rational number: $51.0\overline{2}$Preview
  10. Q55If the common ratio of a G.P. is $\dfrac23$ and sum to infinity is 12, find the first term.Preview
  11. Q56If the first term of the G.P. is 6 and its sum to infinity is $\dfrac{96}{17}$, find the common ratio.Preview
  12. Q57The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15. Find the G.P.Preview
  13. Q58Find $\displaystyle\sum_{r=1}^{\infty}4(0.5)^r$.Preview
  14. Q59Find $\displaystyle\sum_{r=1}^{\infty}\left(-\dfrac13\right)^{r-1}$.Preview
  15. Q60Find $\displaystyle\sum_{r=0}^{\infty}8\left(-\dfrac12\right)^r$.Preview
  16. Q61Find $\displaystyle\sum_{n=1}^{\infty}(0.4)^n$.Preview
  17. Q62The mid points of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of…Preview
  18. Q63A ball is dropped from a height of 10m. It bounces to a height of 6m, then 3.6m and so on. Find the total distance travelled by the ball.Preview
2.3.1

The General term or the nth term of a G.P.

For a G.P. , , , , if is the first term and the common ratio, the th term is (this follows because reaching the th term from the first requires multiplying by exactly times).

2.3.2

Sum of the first n terms of a G.P. (Sn)

Consider the G.P. ; the sum of its first terms is written . Here is the notation for summation and is the running (dummy) variable.

2.4

Sum of infinite terms of a G.P.

Consider a G.P. of positive terms. The sum of the first terms is , . If , grows without bound as , so the infinite sum cannot be found (does not exist).

+EXERCISE 2.413 questions
  1. Q64Verify whether the following sequence is a H.P.: $\dfrac13, \dfrac15, \dfrac17, \dfrac19, \ldots$Free
  2. Q65Verify whether the following sequence is a H.P.: $\dfrac13, \dfrac16, \dfrac1{12}, \dfrac1{24}, \ldots$Free
  3. Q66Verify whether the following sequence is a H.P.: $5, \dfrac{10}{17}, \dfrac{10}{32}, \dfrac{10}{47}, \ldots$Free
  4. Q67Find the $n$th term and hence find the 8th term of the H.P.: $\dfrac12, \dfrac15, \dfrac18, \dfrac1{11}, \ldots$Preview
  5. Q68Find the $n$th term and hence find the 8th term of the H.P.: $\dfrac14, \dfrac16, \dfrac18, \dfrac1{10}, \ldots$Preview
  6. Q69Find the $n$th term and hence find the 8th term of the H.P.: $\dfrac15, \dfrac1{10}, \dfrac1{15}, \dfrac1{20}, \ldots$Preview
  7. Q70Find A.M. of two positive numbers whose G.M. and H.M. are 4 and $\dfrac{16}{5}$ respectively.Preview
  8. Q71Find H.M. of two positive numbers whose A.M. and G.M. are $\dfrac{15}{2}$ and 6.Preview
  9. Q72Find G.M. of two positive numbers whose A.M. and H.M. are 75 and 48.Preview
  10. Q73Insert two numbers between $\dfrac14$ and $\dfrac13$ so that the resulting sequence is a H.P.Preview
  11. Q74Insert two numbers between 1 and $-27$ so that the resulting sequence is a G.P.Preview
  12. Q75If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by $\dfrac{18}{5}$, find the numbers.Preview
  13. Q76Find two numbers whose A.M. exceeds their G.M. by $\dfrac12$ and their H.M. by $\dfrac{25}{26}$.Preview
2.4.1

Expressing recurring decimals as rational numbers

A recurring decimal fraction can always be written as a rational number, e.g. ; this can also be verified using the infinite G.P. sum formula.

2.5

Harmonic Progression (H.P.)

A sequence (with for all ) is called a Harmonic Progression (H.P.) if the reciprocals are in A.P. E.g. (i) are in H.P. because (their reciprocals) are in A.P. (ii) is H.P. because are in A.P.

+EXERCISE 2.57 questions
  1. Q77Find $S_n$ of the following arithmetico-geometric sequence: $2, 4x, 6x^2, 8x^3, 10x^4, \ldots$Free
  2. Q78Find $S_n$ of the following arithmetico-geometric sequence: $1, 4x, 7x^2, 10x^3, 13x^4, \ldots$Free
  3. Q79Find $S_n$ of the following arithmetico-geometric sequence: $1, 2\times3, 3\times9, 4\times27, 5\times81, \ldots$Free
  4. Q80Find $S_n$ of the following arithmetico-geometric sequence: $3, 12, 36, 96, 240, \ldots$Preview
  5. Q81Find the sum to infinity of the following arithmetico-geometric sequence: $1, \dfrac24, \dfrac3{16}, \dfrac4{64}, \ldots$Preview
  6. Q82Find the sum to infinity of the following arithmetico-geometric sequence: $3, \dfrac65, \dfrac9{25}, \dfrac{12}{125}, \dfrac{15}{625}, \ldot…Preview
  7. Q83Find the sum to infinity of the following arithmetico-geometric sequence: $1, -\dfrac43, \dfrac79, -\dfrac{10}{27}, \ldots$Preview
2.6

Types of Means

For two numbers and , three kinds of "mean" (middle value) are defined depending on which progression the triple should form: the Arithmetic Mean makes an A.P.; the Geometric Mean makes a G.P.; and th…

+EXERCISE 2.610 questions
  1. Q84Find $\displaystyle\sum_{r=1}^{n}(r+1)(2r-1)$.Free
  2. Q85Find $\displaystyle\sum_{r=1}^{n}(3r^2-2r+1)$.Free
  3. Q86Find $\displaystyle\sum_{r=1}^{n}\dfrac{1+2+3+\cdots+r}{r}$.Free
  4. Q87Find $\displaystyle\sum_{r=1}^{n}\dfrac{r^2}{1^3+2^3+3^3+\cdots+r^3}$ (source scan of this fraction is corrupted/ambiguous; structure recons…Preview
  5. Q88Find the sum $5\times7+9\times11+13\times15+\cdots$ upto $n$ terms.Preview
  6. Q89Find the sum $2^2+4^2+6^2+8^2+\cdots$ upto $n$ terms.Preview
  7. Q90Find $(70^2-69^2)+(68^2-67^2)+(66^2-65^2)+\cdots+(2^2-1^2)$.Preview
  8. Q91Find the sum $1\times3\times5+3\times5\times7+5\times7\times9+\cdots+(2n-1)(2n+1)(2n+3)$.Preview
  9. Q92If $\dfrac{1\times2+2\times3+3\times4+4\times5+\cdots \text{ upto } n \text{ terms}}{1+2+3+4+\cdots \text{ upto } n \text{ terms}}=\dfrac{10…Preview
  10. Q93If $S_1, S_2$ and $S_3$ are the sums of first $n$ natural numbers, their squares and their cubes respectively, then show that $9S_2^2=S_3(1+…Preview
2.6.1

Arithmetic mean (A.M.)

If and are two numbers, their Arithmetic Mean is . This is precisely the value that makes an A.P., since .

2.6.2

Geometric mean (G.M.)

If and are two numbers of the same sign (both positive or both negative), their Geometric Mean is . This is the value that makes a G.P., since .

2.6.3

Harmonic mean (H.M.)

If and are two numbers, their Harmonic Mean is . This is the value that makes an H.P., since must be in A.P.: .

2.7

Arithmetico-Geometric Progression (A.G.P.)

A sequence in which each term is the product of the corresponding terms of an A.P. and a G.P. is called an Arithmetico-Geometric Progression (A.G.P.). E.g. consider the A.P.

2.7.1

Sum of n terms of A.G.P.

Let . Multiplying by : . Subtracting, the differences between successive bracketed terms all become , leaving a genuine G.P. of terms in , plus the leftover boundary terms: .

2.7.2

Properties of Summation and Standard Results

Properties of Summation: (i) for a nonzero constant ; (ii) ; (iii) ; (iv) for a nonzero constant (a restatement of (i) and (iii) together).

2.8

Power Series

Some functions can be expressed as infinite sums of powers of ; these are called power series. Examples: (1) (2) (3) (4) (5) if then .

More questions

43 Q
+Show 10 questions10 questions
  1. Q94Select the correct answer from the given alternatives. The common ratio for the G.P. $0.12, 0.24, 0.48, \ldots$ is: A) 0.12 B) 0.2 C) 0.02 D…Free
  2. Q95Select the correct answer from the given alternatives. The tenth term of the geometric sequence $\dfrac14, -\dfrac12, 1, -2, \ldots$ is: A)…Free
  3. Q96Select the correct answer from the given alternatives. If for a G.P. $\dfrac{t_6}{t_3}=\dfrac{1458}{54}$ then $r=$? A) 3 B) 2 C) 1 D) $-1$Free
  4. Q97Select the correct answer from the given alternatives. Which term of the geometric progression $1, 2, 4, 8, \ldots$ is 2048? A) 10th B) 11th…Preview
  5. Q98Select the correct answer from the given alternatives. If common ratio of the G.P. is 5, 5th term is 1875, the first term is: A) 3 B) 5 C) 1…Preview
  6. Q99Select the correct answer from the given alternatives. The sum of 3 terms of a G.P. is $\dfrac{21}{4}$ and their product is 1, then the comm…Preview
  7. Q100Select the correct answer from the given alternatives. Sum to infinity of a G.P. $5, -\dfrac52, \dfrac54, -\dfrac58, \dfrac5{16}, \ldots$ is…Preview
  8. Q101Select the correct answer from the given alternatives. The tenth term of H.P. $\dfrac29, \dfrac17, \dfrac2{19}, \dfrac1{12}, \ldots$ is: A)…Preview
  9. Q102Select the correct answer from the given alternatives. Which of the following is not true, where $A, G, H$ are the AM, GM, HM of $a$ and $b$…Preview
  10. Q103Select the correct answer from the given alternatives. The G.M. of two numbers exceeds their H.M. by $\dfrac65$, the A.M. exceeds the G.M. b…Preview
+Show 33 questions33 questions
  1. Q104In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term.Free
  2. Q105Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is $\dfrac23$.Free
  3. Q106For a G.P., $a=\dfrac43$ and $t_7=\dfrac{243}{1024}$, find the value of $r$.Free
  4. Q107For a sequence, if $t_n=\dfrac{5^{n-2}}{7^{n-3}}$, verify whether the sequence is a G.P. If it is a G.P., find its first term and the common…Preview
  5. Q108Find three numbers in G.P. such that their sum is 35 and their product is 1000.Preview
  6. Q109Find five numbers in G.P. such that their product is 243 and sum of second and fourth number is 10.Preview
  7. Q110For a sequence $S_n=4(7^n-1)$ verify that the sequence is a G.P.Preview
  8. Q111Find $2+22+222+2222+\cdots$ upto $n$ terms.Preview
  9. Q112Find the $n$th term of the sequence $0.6, 0.66, 0.666, 0.6666, \ldots$Preview
  10. Q113Find $\displaystyle\sum_{r=1}^{n}(5r^2+4r-3)$.Preview
  11. Q114Find $\displaystyle\sum_{r=1}^{n}r(r-3)(r-2)$.Preview
  12. Q115Find $\displaystyle\sum_{r=1}^{n}\dfrac{1+2+3+\cdots+r}{r^2}$ (source scan of this fraction chain is corrupted/ambiguous; structure reconstr…Preview
  13. Q116Find $\displaystyle\sum_{r=1}^{n}\dfrac{1^3+2^3+3^3+\cdots+r^3}{r}$ (source scan of this fraction chain is corrupted/ambiguous; structure re…Preview
  14. Q117Find $2\times6+4\times9+6\times12+\cdots$ upto $n$ terms.Preview
  15. Q118Find $2\times5\times8+4\times7\times10+6\times9\times12+\cdots$ upto $n$ terms.Preview
  16. Q119Find $\dfrac1{1^2}+\dfrac{1+2}{2^2}+\dfrac{1+2+3}{3^2}+\cdots$ upto $n$ terms (source scan of this fraction chain is corrupted/ambiguous; st…Preview
  17. Q120Find $12^2+13^2+14^2+15^2+\cdots+20^2$.Preview
  18. Q121If $\dfrac{1+2+3+4+5+\cdots \text{ upto } n \text{ terms}}{1\times2+2\times3+3\times4+4\times5+\cdots \text{ upto } n \text{ terms}}=\dfrac{…Preview
  19. Q122Find $(50^2-49^2)+(48^2-47^2)+(46^2-45^2)+\cdots+(2^2-1^2)$.Preview
  20. Q123If $\dfrac{1\times3+2\times5+3\times7+\cdots \text{ upto } n \text{ terms}}{1^2+2^2+3^2+\cdots \text{ upto } n \text{ terms}}=\dfrac59$, fin…Preview
  21. Q124For a G.P. if $t_2=7$, $t_4=1575$, find $a$.Preview
  22. Q125If for a G.P. $t_3=\dfrac13$, $t_6=\dfrac1{81}$, find $r$.Preview
  23. Q126Find $\displaystyle\sum_{r=1}^{n}\dfrac{2}{3^r}$.Preview
  24. Q127Find $k$ so that $k-1, k, k+2$ are consecutive terms of a G.P.Preview
  25. Q128If for a G.P. first term is $(27)^2$ and seventh term is $(8)^2$, find $S_8$.Preview
  26. Q129If $p$th, $q$th and $r$th terms of a G.P. are $x, y, z$ respectively, find the value of $x^{q-r}\cdot y^{r-p}\cdot z^{p-q}$.Preview
  27. Q130Which 2 terms are inserted between 5 and 40 so that the resulting sequence is G.P.?Preview
  28. Q131If $p, q, r$ are in G.P. and $p^{1/x}=q^{1/y}=r^{1/z}$, verify whether $x, y, z$ are in A.P. or G.P. or neither.Preview
  29. Q132If $a, b, c$ are in G.P. and $ax^2+2bx+c=0$ and $px^2+2qx+r=0$ have a common root, then verify that $pb^2-2qba+ra^2=0$.Preview
  30. Q133If $p, q, r, s$ are in G.P., show that $(p^2+q^2+r^2)(q^2+r^2+s^2)=(pq+qr+rs)^2$.Preview
  31. Q134If $p, q, r, s$ are in G.P., show that $(p^n+q^n), (q^n+r^n), (r^n+s^n)$ are also in G.P.Preview
  32. Q135Find the coefficient of $x^6$ in the expansion of $e^{2x}$ using series expansion.Preview
  33. Q136Find the sum of infinite terms of $1+\dfrac45+\dfrac7{25}+\dfrac{10}{125}+\dfrac{13}{625}+\cdots$Preview