# NCERT Solutions for Class 11 Maths Maths Chapter 9

## Sequences and Series Class 11

### Exercise 9.1 : Solutions of Questions on Page Number : 180

Q1 :

Write the first five terms of the sequences whose nth term is  Substituting n = 1, 2, 3, 4, and 5, we obtain Therefore, the required terms are 3, 8, 15, 24, and 35.

Q2 :

Write the first five terms of the sequences whose nth term is  Substituting n = 1, 2, 3, 4, 5, we obtain Therefore, the required terms are .

Q3 :

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

Q4 :

Write the first five terms of the sequences whose nth term is Q5 :

Write the first five terms of the sequences whose nth term is Q6 :

Write the first five terms of the sequences whose nth term is Q7 :

Find the 17th term in the following sequence whose nth term is Q8 :

Find the 7th term in the following sequence whose nth term is Q9 :

Find the 9th term in the following sequence whose nth term is Q10 :

Find the 20th term in the following sequence whose nth term is Q11 :

Write the first five terms of the following sequence and obtain the corresponding series: Q12 :

Write the first five terms of the following sequence and obtain the corresponding series: Q13 :

Write the first five terms of the following sequence and obtain the corresponding series: Q14 :

The Fibonacci sequence is defined by Find ### Exercise 9.2 : Solutions of Questions on Page Number : 185

Q1 :

Find the sum of odd integers from 1 to 2001.

Q2 :

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

Q3 :

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.

Q4 :

How many terms of the A.P. are needed to give the sum â€“25?

Q5 :

In an A.P., if pth term is and qth term is , prove that the sum of first pq terms is Q6 :

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

Q7 :

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

Q8 :

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

Q9 :

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

Q10 :

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.

Q11 :

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

Prove that Q12 :

The ratio of the sums of m and n terms of an A.P. is m2: n2. Show that the ratio of mth and nth term is (2m - 1): (2n - 1).

Q13 :

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

Q14 :

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

Q15 :

If is the A.M. between a and b, then find the value of n.

Q16 :

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.

Q17 :

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?

Q18 :

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 : Solutions of Questions on Page Number : 192

Q1 :

Find the 20th and nthterms of the G.P. Q2 :

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

Q3 :

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

Q4 :

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

Q5 :

Which term of the following sequences:

(a) (b) (c) Q6 :

For what values of x, the numbers are in G.P?

Q7 :

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

Q8 :

Find the sum to n terms in the geometric progression Q9 :

Find the sum to n terms in the geometric progression Q10 :

Find the sum to n terms in the geometric progression Q11 :

Evaluate Q12 :

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

Q13 :

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

Q14 :

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.

Q15 :

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

Q16 :

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

Q17 :

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.

Q18 :

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

Q19 :

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

Q20 :

Show that the products of the corresponding terms of the sequences form a G.P, and find the common ratio.

Q21 :

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.

Q22 :

If the terms of a G.P. are a, b and c, respectively. Prove that Q23 :

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

Q24 :

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from .

Q25 :

If a, b, c and d are in G.P. show that .

Q26 :

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

Q27 :

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

Q28 :

The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio .

Q29 :

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are .

Q30 :

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?

Q31 :

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

Q32 :

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 : Solutions of Questions on Page Number : 196

Q1 :

Find the sum to n terms of the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …

Q2 :

Find the sum to n terms of the series 1 x 2 x 3 + 2 x 3 x 4 + 3 x 4 x 5 + …

Q3 :

Find the sum to n terms of the series 3 x 12 + 5 x 22 + 7 x 32 + …

Q4 :

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

3 x 12 + 5 x 22 + 7 x 32 + …

Q5 :

Find the sum to n terms of the series Q6 :

Find the sum to n terms of the series Q7 :

Find the sum to n terms of the series 3 x 8 + 6 x 11 + 9 x 14 +…

Q8 :

Find the sum to n terms of the series 12 + (12 + 22) + (12 + 22 + 32) + …

Q9 :

Find the sum to n terms of the series whose nth term is given by n (n + 1) (n + 4).

Q10 :

Find the sum to n terms of the series whose nth terms is given by n2 + 2n

Q11 :

Find the sum to n terms of the series whose nth terms is given by (2n - 1)2

### Exercise Miscellaneous : Solutions of Questions on Page Number : 199

Q1 :

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

Q2 :

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

Q3 :

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

Q4 :

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

Q5 :

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

Q6 :

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

Q7 :

If f is a function satisfying such that , find the value of n.

Q8 :

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.

Q9 :

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.

Q10 :

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.

Q11 :

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.

Q12 :

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.

Q13 :

If , then show that a, b, c and d are in G.P.

Q14 :

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

Q15 :

The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that Q16 :

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

Q17 :

If a, b,c, d are in G.P, prove that are in G.P.

Q18 :

If a and b are the roots of are roots of , where a, b, c, d, form a G.P. Prove that (q + p): (q â€“ p) = 17:15.

Q19 :

The ratio of the A.M and G.M. of two positive numbers a and b, is m:n. Show that .

Q20 :

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

Q21 :

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

(i) 5 + 55 + 555 + … (ii) .6 +.66 +. 666 +…

Q22 :

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

Q23 :

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

Q24 :

If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that Q25 :

Find the sum of the following series up to n terms: Q26 :

Show that Q27 :

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?

Q28 :

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?

Q29 :

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.

Q30 :

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.

Q31 :

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.

Q32 :

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.