cbse ncert 10th mathematics
chapter 5 arithmetic progressions, exercise 5.2
5.Find the number of terms in each of the following APs :
7,13,19, ....,205
a =7
d=t2-t1 =13-7 =6
last term L=205
either use the formula
n=[(L-a)/d]+1 = [(205-7)/6]+1 =33+1=34 terms
OR
using the formula for the nth term of an arithmetic progression AP
tn= a+ (n-1)d with tn=205 and n=?
205 = 7 +(n-1)(6)
(205-7)/6 =( n-1)
33=n-1
n=33+1 =34
(ii) 18, [15+(1/2)],13, . . . ,(-47)
a = 18
change the mixed fraction for the second term
t2=(31/2)
d= (31/2) -18 =(-5/2)
last term L=(-47)
either use direct formula
n=[(L-a)/d]+1 = [{(-47)-18 }/ {-5/2} ]+1
n= [(-65)*(-2/5)]+1
n=26+1=27
OR
tn=(-47) with n=?
using the formula for the nth term of an arithmetic progression AP
tn= a+ (n-1)d
(-47) = 18 +(n-1)(-5/2)
(-47-18) / (-5/2) =(n-1)
n= [(-65)*(-2/5)]+1
n=26+1=27
4. Which term of the AP : 3, 8, 13, 18, . . . ,is 78?
a=3
d=t2-t1=8-3=5
tn=78 with n=?
using the formula for the nth term of an arithmetic progression AP
tn= a+ (n-1)d
78 =3 +(n-1)(5)
78-3=(n-1)(5)
75=(n-1)(5)
75/5=(n-1)
15+1=n
n=16
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ncert cbse 10th mathematics
chapter 5 arithmetic progressions
exercise 5.4 optional exercise
Which term of the AP : 121, 117, 113, . . ., is its first negative term?
2. The sum of the third and the seventh terms of an AP is 6 and their
product is 8. Find the sum of first sixteen terms of the AP.
3. A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and last rungs are [ 2 and(1/2) ]m apart, what is the length of the wood required for the rungs?
4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x.
solution5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of (1/4) m and a tread of (1/2)m. Calculate the total volume of concrete required to build the terrace.
chapter 5 arithmetic progressions, exercise 5.3
exercise 5.3
20. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato,and the other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
19.
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on . In how many rows are the 200 logs placed and how many logs are in the top row?
18. A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, . . . What is the total length of such a spiral made up of thirteen consecutive semicircles.
17. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
16. A sum of Rs.700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs.20 less than its preceding prize, find the value of each of the prizes.
15. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs.300 for the third day, etc., the penalty for each succeeding day being Rs.50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
14. Find the sum of the odd numbers between 0 and 50.
13. Find the sum of the first 15 multiples of 8.
12. Find the sum of the first 40 positive integers divisible by 6.
11.If the sum of the first n terms of an AP is 4n –(n^2) , what is the first term (that is S1 )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.
10. Show that a1 , a2 , . . ., an , . . . form an AP where a n is defined as below :
an = 3 + 4n
(ii) an = 9 – 5n
Also find the sum of the first 15 terms in each case.
9. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.
8. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
7. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.
6. The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
5. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
4. How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?
3.
given a = 5, d = 3, an = 50, find n and Sn
(ii) given a = 7, a13 = 35, find d and S13
(iii) given a(12) = 37, d = 3, find a and S(12 )
(iv) given a3 = 15, S(10) = 125, find d and a(10)
(v) given d = 5, S9 = 75, find a and a9 .
(vi) given a = 2, d = 8, Sn = 90, find n and an .
vii) given a = 8, an = 62, Sn = 210, find n and d
(vii) given an = 4, d = 2, Sn = –14, find n and a.
ix) given a = 3, n = 8, S = 192, find d.
(x) given L= 28, S = 144, and there are total 9 terms. Find a.
find the sums given below :
7 + [10 +(1/2) ] +14 + ...+84
(ii) 34 + 32 + 30 + . . . + 10
(iii) –5 + (–8) + (–11) + . . . + (–230)
Find the sum of the following APs:
2, 7, 12, . . ., to 10 terms.
(iii) 0.6, 1.7, 2.8, . . ., to 100 terms
(iv) (1/15) +(1/12) +(1/10) + .... 11terms
Aruna saved Rs. 5 in the first week of a year and then increased her weekly savings by Rs. 1.75. If in the nth week, her weekly savings become Rs.20.75, find n.
exercise 5.2
19. Subba Rao started work in 1995 at an annual salary of Rs. 5000 and received an increment of Rs.200 each year. In which year did his income reach Rs.7000?
18. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.
17.Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253.
16. Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.
15. For what value of n, are the nth terms of two APs: 63, 65, 67, . . .
and 3, 10, 17, . . . equal?
14. How many multiples of 4 lie between 10 and 250?
13.How many three-digit numbers are divisible by 7?
11. Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?
10. The 17th term of an AP exceeds its 10th term by 7. Find the common difference.
9. If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP is zero?
8. An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.
7. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
6. Check whether (-150) is a term of the AP : 11, 8, 5, 2 . . .
7. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
5. Find the number of terms in each of the following APs :
7,13,19, ....,205
(ii) 18, [15+(1/2)],13, . . . ,(-47)
4. Which term of the AP : 3, 8, 13, 18, . . . ,is 78?
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