Class 11 Maths NCERT Solutions For Chapter 9 Sequences and Series

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Chapter 9 Sequences and Series

Download NCERT Solutions for Class 11 Mathematics

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EXERCISE 9.1

Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are:

Question 1.an = n (n + 2)

Question 2.an = 1 n n +

Question 3.an = 2n

Question 4.an = 2 3 6 n −

Question 5.an = (–1)n–1 5n+1

Question 6.an 2 5 4 n n + = . Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are:

Question 7.an = 4n – 3; a17, a24

Question 8.an = 2 7 ; 2n n a

Question 9.an = (–1)n – 1n3; a9 Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the corresponding series:

Question 10.20 ( –2); n 3 a n n a n =

EXERCISE 9.2

Question 1.Find the sum of odd integers from 1 to 2001.

Question 2.Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.

Question 3.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 –11 2.

Question 4.How many terms of the A.P. – 6, 11 2 − , – 5, … are needed to give the sum –25?

Question 5.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.

Question 6.If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 11 7 Find the last term. Find the sum to n terms of the A.P., whose kth term is 5k +1

Question 8.If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.

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

Question 10.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. 1

Question 1.Sum of the first p, q and r terms of an A.P are. a, b and c, respectively. Prove that a (q r) b (r p) c ( p q) 0 p q r − + − + − = 1

Question 2.The ratio of the sums of m and n terms of an A.P. is m2 : n 2.Show that the ratio of mth and nth term is (2m – 1) : (2n – 1). 1

Question 3.If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m. 1

Question 4.Insert five numbers between 8 and 26 such that the resulting sequence is an A.P. 1

Question 5.If 1 1 n n n n a b a − b − + + is the A.M. between a and b, then find the value of n.

Question 6.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.

Question 7.A man starts repaying a loan as first instalment of Rs. 100. If he increases the instalment by Rs 5 every month, what amount he will pay in the 30th instalment?

Question 8.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.

EXERCISE 9.3

Question 1.Find the 20th and nth terms of the G.P. 5 5 5 2 4 8 , , , ...

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

Question 3.The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.

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

Question 5.Which term of the following sequences: (a) 2,2 2,4,... is 128 ? (b) 3,3,3 3,...is729 ? (c) 1 1 1 is 1 3 9 27 19683 , , ,... ?

Question 6.For what values of x, the numbers 2 2 7 7 – ,x,– are in G.P.? Find the sum to indicated number of terms in each of the geometric progressions in Exercises 7 to 10:

Question 7.0.15, 0.015, 0.0015, ... 20 terms.

Question 8.7 , 21 , 3 7 , ... n terms.

Question 9.1, – a, a2, – a3, ... n terms (if a ≠ – 1).

Question 10.x3, x5, x7, ... n terms (if x ≠ ± 1).

Question 1.Evaluate 11 1 (2 3k ) k = Σ + . 1

Question 2.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.

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

Question 4.The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 12 8. Determine the first term, the common ratio and the sum to n terms of the G.P.

Question 5.Given a G.P. with a = 729 and 7th term 64, determine S 7.

Question 6.Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.

Question 7.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.

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

Question 9.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 20.

Show that the products of the corresponding terms of the sequences a, ar, ar2, …arn – 1 and A, AR, AR2, … ARn – 1 form a G.P, and find the common ratio.

Question 1.Find four numbers forming a geometric progression in which the third term is greater than the first term by 9, and the second term is greater than the 4th by 18.

Question 2.If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq – r br – pcP – q =1.

Question 3.If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.

Question 4.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 rn .

Question 5.If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2

Question 6.Insert two number between 3 and 81 so that the resulting sequence is G.P. 2

Question 7.Find the value of n so that a b a b n n n n + + + + 1 1 may be the geometric mean between a and b.

Question 8.The sum of two numbers is 6 times their geometric means, show that numbers are in the ratio (3+ 2 2 ): (3−2 2).

Question 9.If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are A A G A G ( )( ) ± + −. 30.

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 nth hour ?

Question 1.What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

Question 2.If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation

EXERCISE 9.4

Find the sum to n terms of each of the series in Exercises 1 to 7.

Question 1. 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 +...

Question 2. 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + ...

Question 3. 3 × 12 + 5 × 22 + 7 × 32 + ...

Question 4. 1 1 1 1 2 2 3 3 4 + + + × × × ...

Question 5. 52 + 62 + 72 + ... + 202

Question 6 .3 × 8 + 6 × 11 + 9 × 14 + ...

Question 7. 12 + (12 + 22) + (12 + 22 + 32) + ...

Find the sum to n terms of the series in Exercises 8 to 10 whose nth terms is given by

Question 8.n (n+1) (n+4).

Question 9.n2 + 2n

Question 10.(2n – 1)2

Miscellaneous Exercise On Chapter 9

Question 1.Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.

Question 2.If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

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

Question 4.Find the sum of all numbers between 200 and 400 which are divisible by 7.

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

Question 6.Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

Question 7.If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and 1 ( ) 120 n x f x = Σ = , find the value of n.

Question 8.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

Question 9.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.

Question 10.The sum of three numbers in G.P. is 5 6.If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression.

Find the numbers.

Question 1.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.

Question 2.The sum of the first four terms of an A.P. is 56.The sum of the last four terms is 12. If its first term is 11, then find the number of terms.

Question 3.If a bx a bx b cx b cx c dx c dx x + − = + − = + − ( ≠ 0) , then show that a, b, c and d are in G.P.

Question 4. 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.

Question 5.The pth, qth and rth terms of an A.P. are a, b, c, respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0

Question 6.If a 1 1 ,b 1 1 ,c 1 1 b c c a a b are in A.P., prove that a, b, c are in A.P.

Question 7.If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.

Question 8.If a and b are the roots of x2 – 3x + p = 0 and c, d are roots of x2 – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17:15.

Question 9.The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a :b = (m + m2 – n2 ): (m – m2 – n2 ) 20. If a, b, c are in A.P.; b, c, d are in G.P. and 1 , 1 ,1 c d e are in A.P. prove that a, c, e are in G.P.

Question 10.Find the sum of the following series up to n terms: (i) 5 + 55 +555 + … (ii) .6 +. 66 +. 666+…

Question 11.Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms.

Question 12.Find the sum of the first n terms of the series: 3+ 7 +13 +21 +31 +…

Question 13.If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that 9 22 S = S3 (1 + 8S1).

Question 14 .Find the sum of the following series up to n terms: 13 13 22 13 23 33 1 1 3 1 3 5... + + + + + + + + +

Question 15.Show that 2 2 2 2 2 2 1 2 2 3 ( 1) 3 5 1 2 2 3 ( 1) 3 1 ... n n n ... n n n × + × + + × + + = × + × + + × + + .

Question 16.A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?

Question 17.Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

Question 18.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.

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