Solution of Chapter 5. Arithemetic Progressions (NCERT - Mathematics Book)

In which of the following situations, does the list of numbers involved make an arithmeticprogression, and why?

(i) The taxi fare after each km when the fare is Rs 15 for the first km and Rs 8 for each additional km.

(ii) The amount of air present in a cylinder when a vacuum pump removesof the air remaining in the cylinder at a time.

(iii) The cost of digging a well after every metre of digging, when it costs Rs 150 forthe first metre and rises by Rs 50 for each subsequent metre.

(iv) The amount of money in the account every year, when Rs 10000 is deposited atcompound interest at 8 % per annum

Write first four terms of the AP, when the first term a and the common difference d aregiven as follows:

(i)

(ii)

(iii)

(iv)

(v)

For the following APs, write the first term and the common difference:

(i)

(ii)

(iii)

(iv)

Which of the following are APs ? If they form an AP, find the common difference d andwrite three more terms.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix) 1, 3, 9, 27,…

(x)

(xi)

(xii)

(xiii)

(xiv)

Fill in the blanks in the following table, given that a is the first term, d the commondifference and the nth term of the AP:

Choose the correct choice in the following and justify:

(i) 30th term of the AP: 10, 7, 4, . . . , is

A. 97 B. 77

C. –77 D. – 87

(ii) 11th term of the AP: is

A. 28 B. 22

C. –38 D. – 48

In the following APs, find the missing terms in the boxes :

(i)

(ii)

(iii)

(iv)

(v)

Which term of the AP : 3, 8, 13, 18, . . . ,is 78?

Find the number of terms in each of the following APs :

(i) 7, 13, 19, . . . , 205

(ii)

Check whether – 150 is a term of the AP : 11, 8, 5, 2 . . .

Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73

An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29^thterm.

If the 3rd and the 9th terms of an AP are 4 and – 8 respectively, which term of this AP iszero?

The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

Which term of the AP : 3, 15, 27, 39, . . . will be 132 more than its 54th term?

Two APs have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

How many three-digit numbers are divisible by 7?

How many multiples of 4 lie between 10 and 250?

For what value of n, are the nth terms of two APs: 63, 65, 67, . . . and 3, 10, 17, . . . equal?

Determine the AP whose third term is 16 and the 7th term exceeds the 5th term by 12.

Find the 20th term from the last term of the AP : 3, 8, 13, . . ., 253

The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is44. Find the first three terms of the AP.

SubbaRao started work in 1995 at an annual salary of Rs 5000 and received an incrementof Rs 200 each year. In which year did his income reach Rs 7000?

Ramkali saved Rs 5 in the first week of a year and then increased her weekly savings byRs 1.75. If in the nth week, her weekly savings become Rs 20.75, find n.

Find the sum of the following APs:

(i) 2, 7, 12, . . ., to 10 terms.

(ii) –37, –33, –29, . . ., to 12 terms.

(iii) 0.6, 1.7, 2.8, . . ., to 100 terms.

(iv)

Find the sums given below :

(i)

(ii)

(iii)

In an AP:

(i) givenfind and

(ii) givenfind and

(iii) given find and

(iv) givenfind and

(v) givenfind a and

(vi) givenfind and

(vii) given find and

(viii) given find and

(ix) given find

(x) givenand there are total 9 terms. Find

How many terms of the AP : 9, 17, 25, . . . must be taken to give a sum of 636?

The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of termsand the common difference.

The first and the last terms of an AP are 17 and 350 respectively. If the common differenceis 9, how many terms are there and what is their sum?

Find the sum of first 22 terms of an AP in which and 22nd term is 149.

Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18respectively.

If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum offirst n terms.

Show that form an AP where is defined as below :

(i) (ii)

Also find the sum of the first 15 terms in each case.

If the sum of the first n terms of an AP is what is the first term (that is S1)?

Whatis the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th andthe nth terms.

Find the sum of the first 40 positive integers divisible by 6

Find the sum of the first 15 multiples of 8

Find the sum of the odd numbers between 0 and 50

A contract on construction job specifies a penalty for delay of completion beyond acertain 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 thepreceding day. How much money the contractor has to pay as penalty, if he has delayedthe work by 30 days?

A sum of Rs 700 is to be used to give seven cash prizes to students of a school for theiroverall academic performance. If each prize is Rs 20 less than its preceding prize, find thevalue of each of the prizes.

In a school, students thought of planting trees in and around the school to reduce airpollution. It was decided that the number of trees, that each section of each class willplant, will be the same as the class, in which they are studying, e.g., a section of Class Iwill plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There arethree sections of each class. How many trees will be planted by the students?

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, . . . as shown inFig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take

[Hint : Length of successive semicircles is with centres at A, B, A, B, . . .,respectively.]

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 (see Fig. 5.5). In how may rows are the 200 logs placedand how many logs are in the top row?

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 theline (see Fig. 5.6).

A competitor starts from the bucket, picks up the nearest potato, runs back with it, dropsit in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, andshe continues in the same way until all the potatoes are in the bucket. What is the totaldistance the competitor has to run?

[Hint : To pick up the first potato and the second potato, the total distance (in metres)run by a competitor is

Which term of the AP: 121, 117, 113, . . ., isits first negative term?

[Hint: Find n for ]

The sum of the third and the seventh termsof an AP is 6 and their product is 8. Findthe sum of first sixteen terms of the AP.

A ladder has rungs 25 cm apart.(see Fig. 5.7). The rungs decreaseuniformly in length from 45 cm at thebottom to 25 cm at the top. If the top andthe bottom rungs arem apart, what isthe length of the wood required for therungs?

[Hint : Number of rungs

The houses of a row are numbered consecutively from 1 to 49. Show that there is a valueof x such that the sum of the numbers of the houses preceding the house numbered x isequal to the sum of the numbers of the houses following it. Find this value of x.

[Hint :]

A small terrace at a football ground comprises of 15 steps each of which is 50 m long andbuilt of solid concrete.

Each step has a rise ofm and a tread ofm. (see Fig. 5.8). Calculate the total volumeof concrete required to build the terrace.

[Hint : Volume of concrete required to build the first step ]