shaalaa.com
S

NCERT solutions Mathematics Textbook for Class 11 chapter 9 Sequences and Series

Account
User


Login
Register


      Forgot password?
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Chapter 9 - Sequences and Series

Pages 180 - 181

Write the first five terms of the sequences whose nth term is `a_n = n(n+2)`

 
Q 1 | Page 180 |

Write the first five terms of the sequences whose nth term is  `a_n = n/(n + 1)`

Q 2 | Page 180 |

Write the first five terms of the sequences whose nth term is an = 2n

Q 3 | Page 180 |

Write the first five terms of the sequences whose nth term is  `a_n = (2n -3)/6`

Q 4 | Page 180 |

Write the first five terms of the sequences whose nth term is  `a_n = (-1)^(n-1) 5^(n+1)`

Q 5 | Page 180 |

Write the first five terms of the sequences whose nth term is  `a_n = n (n^2 + 5)/4`

Q 6 | Page 180 |

Find the 17th term in the following sequence whose nth term is an = 4n – 3; `a_17 , a_24`

Q 7 | Page 180 |

Find the indicated terms in each of the sequences whose nth terms are `a_n = n^2/2^n`; `a_7`

Q 8 | Page 180 |

Find the 9th term in the following sequence whose nth term is `a_n = (–1)^(n – 1) n^3; a_9`

Q 9 | Page 180 |

Find the 20th term in the following sequence whose nth term is `a_n = (n(n-2))/(n+3)` ;`a_20`

Q 10 | Page 180 |

Write the first five terms of the following sequence and obtain the corresponding series:

`a_1 = 3, a_n = 3a_( n-1) + 2 for all n > 1`

Q 11 | Page 181 |

Write the first five terms of the following sequence and obtain the corresponding series:   `a_1 = -1, a_n = (a_(n-1))/n , n >= 2`

Q 12 | Page 181 |

Write the first five terms of the following sequence and obtain the corresponding series: 

`a_1 = a_2 = 2, a_n = a_(n-1) -1, n > 2`

Q 13 | Page 181 |

The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2 , n > 2.

Find `a_(n+1)/a_n`, for n = 1, 2, 3, 4, 5

 

Q 14 | Page 181 |

Pages 185 - 186

Find the sum of odd integers from 1 to 2001.

Q 1 | Page 185 |

Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

Q 2 | Page 185 |

In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20th term is –112.

Q 3 | Page 185 |

How many terms of the A.P.  -6 , `-11/2` , -5... are needed to give the sum –25?

Q 4 | Page 185 |

In an A.P., if pth term is 1/q and qth term is 1/p,  prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`

Q 5 | Page 185 |

If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term

Q 6 | Page 185 |

Find the sum to n terms of the A.P., whose kth term is 5k + 1.

Q 7 | Page 185 |

If the sum of n terms of an A.P. is (pn qn2), where p and q are constants, find the common difference.

Q 8 | Page 185 |

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4: 9n + 6. Find the ratio of their 18th terms

Q 9 | Page 185 |

If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Q 10 | Page 185 |

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`

Q 11 | Page 185 |

The ratio of the sums of m and n terms of an A.P. is m2n2. Show that the ratio of mth and nthterm is (2m – 1): (2n – 1)

Q 12 | Page 185 |

If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.

Q 13 | Page 185 |

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Q 14 | Page 185 |

if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.

Q 15 | Page 185 |

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)th numbers is 5:9. Find the value of m.

Q 16 | Page 185 |

A man starts repaying a loan as first installment of Rs. 100. If he increases the installment by Rs 5 every month, what amount he will pay in the 30th installment?

Q 17 | Page 186 |

The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon.

Q 18 | Page 186 |

Pages 192 - 193

Find the 20th and nthterms of the G.P. `5/2, 5/4 , 5/8,...`

Q 1 | Page 192 |

Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.

Q 2 | Page 192 |

The 5th, 8th and 11th terms of a G.P. are pq and s, respectively. Show that q2 = ps.

Q 3 | Page 192 |

The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.

Q 4 | Page 192 |

Which term of the following sequences: 

`2, 2sqrt2, 4,.... is 128`

Q 5.1 | Page 192 |

Which term of the following sequences:

`sqrt3, 3, 3sqrt3, .... is 729`?

Q 5.2 | Page 192 |

Which term of the following sequences:

`1/3, 1/9, 1/27, .... is 1/19683` ?

Q 5.3 | Page 192 |

For what values of x, the numbers  `2/7, x, -7/2` are in G.P?

Q 6 | Page 192 |

Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 …

Q 7 | Page 192 |

Find the sum to n terms in the geometric progression `sqrt7, sqrt21,3sqrt7...`

Q 8 | Page 192 |

Find the sum to n terms in the geometric progression 1, – a, a2, – a3, ... n terms (if a ≠ – 1).

Q 9 | Page 192 |

Find the sum to n terms in the geometric progression x3, x5, x7, ... n terms (if x ≠ ± 1).

Q 10 | Page 192 |

Evaluate `sum_(k=1)^11 (2+3^k )`

Q 11 | Page 192 |

The sum of first three terms of a G.P. is  `39/10` and their product is 1. Find the common ratio and the terms.

Q 12 | Page 192 |

How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?

Q 13 | Page 192 |

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.

Q 14 | Page 192 |

Given a G.P. with a = 729 and 7th term 64, determine S7.

Q 15 | Page 192 |

Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.

Q 16 | Page 192 |

If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.

Q 17 | Page 192 |

Find the sum to n terms of the sequence, 8, 88, 888, 8888…

Q 18 | Page 193 |

Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, 1/2

Q 19 | Page 193 |

Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … `AR^(n-1)` form a G.P, and find the common ratio

Q 20 | Page 193 |

Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.

Q 21 | Page 193 |

If the `p^(th), q^(th) and r^(th)` terms of a G.P. are a, b and c, respectively. Prove that `a^(q - r) b^(r-p) c^(p-q) = 1`

Q 22 | Page 193 |

If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of terms, prove that P2 = (ab)n.

Q 23 | Page 193 |

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from `(n + 1)^(th) " to "(2n)^(th) " term is " 1/r^n`.

 
Q 24 | Page 193 |

If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 .

Q 25 | Page 193 |

Insert two numbers between 3 and 81 so that the resulting sequence is G.P.

Q 26 | Page 193 |

Find the value of n so that  `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.

Q 27 | Page 193 |

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio `3 + 2sqrt2) ":" (3 - 2sqrt2)`

Q 28 | Page 193 |

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are `A+- sqrt((A+G)(A-G))`

Q 29 | Page 193 |

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nthhour?

Q 30 | Page 193 |

What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

Q 31 | Page 193 |

If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.

Q 32 | Page 193 |

Pages 199 - 200

Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

Q 1 | Page 199 |

If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

Q 2 | Page 199 |

Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that S3 = 3 (S2– S1)

Q 3 | Page 199 |

Find the sum of all numbers between 200 and 400 which are divisible by 7.

Q 4 | Page 199 |

Find the sum of integers from 1 to 100 that are divisible by 2 or 5.

Q 5 | Page 199 |

Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

Q 6 | Page 199 |

If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.

Q 7 | Page 199 |

The sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.

Q 8 | Page 199 |

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Q 9 | Page 199 |

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Q 10 | Page 199 |

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Q 11 | Page 199 |

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

Q 12 | Page 199 |

if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that abc and d are in G.P.

Q 13 | Page 199 |

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P2Rn = Sn

Q 14 | Page 199 |

The pthqth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0

Q 15 | Page 199 |

if `a(1/b + 1/c), b(1/c+1/a), c(1/a+1/b)` are in A.P., prove that a, b, c are in A.P.

Q 16 | Page 199 |

If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.

Q 17 | Page 199 |

If a and are the roots of are roots of x2 – 3x + p = 0 , and c, d are roots of x2 – 12x + q = 0, where a, b, cd, form a G.P. Prove that (q + p): (q – p) = 17:15.

Q 18 | Page 199 |

The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that `a:b = (m + sqrt(n^2 - n^2)):(m - sqrt(m^2 - n^2))`.

Q 19 | Page 200 |

If a, b, c are in A.P,; b, c, d are in G.P and ` 1/c, 1/d,1/e` are in A.P. prove that a, c, e are in G.P.

 
Q 20 | Page 200 |

Find the sum of the following series up to n terms:

5 + 55 + 555 + …

Q 21.1 | Page 200 |

Find the sum of the following series up to n terms:

.6 +.66 +. 666 +…

Q 21.2 | Page 200 |

Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + … + n terms.

Q 22 | Page 200 |

Find the sum of the first n terms of the series: 3 + 7 + 13 + 21 + 31 + …

Q 23 | Page 200 |

If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that  `9S_2^2 = S_3(1 + 8S_1)`

Q 24 | Page 200 |

Find the sum of the following series up to n terms `1^3/1 + (1^3 + 2^3)/(1+3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) +...`

Q 25 | Page 200 |

Show that  `(1xx2^2 + 2xx3^2 + ...+nxx(n+1)^2)/(1^2 xx 2 + 2^2 xx3 + ... + n^2xx (n+1))` = `(3n + 5)/(3n + 1)`

Q 26 | Page 200 |

A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?

Q 27 | Page 200 |

Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

Q 28 | Page 200 |

A person 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 instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.

Q 29 | Page 200 |

A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.

Q 30 | Page 200 |

A manufacturer reckons that the value of a machine, which costs him Rs 15625, will depreciate each year by 20%. Find the estimated value at the end of 5 years.

Q 31 | Page 200 |

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

Q 32 | Page 200 |

Where to buy

145
-
242
-
S