Method to write an AP, when first term and common difference are given:

The nth term of an AP is 7 – 4n, then its common difference is

A .  – 3

B .  3

C .   4

D .  – 4

An A.P is defined by an = 4n + 5 , then write the sequence.

Write the first three terms of the AP, when ‘a’ and ‘d’ are as given below:

i. a = 1/2, d = −1/6

ii. a = −5, d = −3

iii. a = √2, d = 1/√2

Find the first 5 terms of the sequence defined by an = (−1)n – 1 × 2n and check whether the sequence is in AP?

Which is the next term of the AP: √2, √8, √18, √32,....?

For what value of k will k + 9, 2k − 1, and 2k + 7 are consecutive terms of an AP.

nth Term of an AP:

Type I: problem based on finding nth term when sequence of an AP is given.

Type II: Problems based on finding n when nth term and AP are given.

Find the 10th term of the AP: 2, 7, 12, . . .

Which term of the AP: 21, 18, 15, . . . is –81? Also, is any term 0? Give reason for your answer.

Check whether 301 is a term of the list of numbers 5, 11, 17, 23, . . .

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

[Hint: Find n for an < 0]

How many two-digit numbers are divisible by 3?

If m times the mth term of an A.P is equal to n times its nth term, show that the (m + n)th term of the AP is zero.

nth Term of an AP:

• nth Term of an AP from the End(Last term):

• Middle Term

• problems based on finding the AP and nth term when its two terms are given

• Word problem

Determine the AP whose 3rd term is 5 and the 7th term is 9.

Find the middle term of the A.P. 213, 205, 197,.... 37.

Find the 11th term from the last term (towards the first term)of the AP: 10, 7, 4, . . . ,– 62.

A sum of ₹1000 is invested at 8% simple interest per year. Calculate the interest at the end of each year. Do these interests form an AP? If so, find the interest at the end of 30 years making use of this fact.

In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?

Sum of First n Terms of an AP and Arithmetic Mean:

Find the sum of the first 22 terms of the AP: 8, 3, –2 . . .

Find the sum of :

(i) the first 1000 positive integers

(ii) the first n positive integers

Solve the equation: 1 + 4 + 7 + 10 + .....+ x = 287.

Find the sum of first 24 terms of the list of numbers whose nth term is given by an = 3 + 2n?

The sum of the first terms of an A.P. is given by Sn = 2n2 + 3n. Find the sixteenth term of the A.P.

Sum of First n Terms of an AP – Word problems:

If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.

How many terms of the AP: 24, 21, 18 . . .must be taken so that their sum is 78?

Find the sum of the first 15 multiples of 8.

A manufacturer of TV sets produced 600 sets in the third year and 700 sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year, find :

(i) the production in the 1st year

(ii) the production in the 10th year

(iii) the total production in first 7 years

A sum of ₹700 is to be used in give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of the prizes.

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

(i) The tax if are after each km when the fare is ₹15 for the first km and ₹8 for each additional km.

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

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

(iv)The amount of money in the account every year, when ₹ 10,000 is deposited at compound Interest at

8%perannum.

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

i. a = 10, d = 10

ii. a=-2,d = 0

iii. a= 4,d =-3

iv. a =-1,d = 1/2

v. a =-1.25, d =-0.25

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

(i). 3,1,–1,–3, ….…

(ii). –5, –1, 3, 7, ……..

(iii).  1/3, 5/3, 9/3, 13/3…….

(iv). 0.6, 1.7, 2.8, 3.9 ...

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

i. 2, 4, 8, 16...

ii. 2, 5/2, 3, 7/2 ….

iii. −1.2, −3.2, −5.2, −7.2...

iv. −10, −6, −2, 2, ...

v. 3, 3+√2, 3+2√2, 3+3√2, …..

vi. 0.2,0.22,0.222,0.2222...

vii. 0, −4, −8, −12...

viii. -1/2,-1/2, - 1/2, - 1/2…..

ix. 1, 3, 9, 27...

x. a, 2a, 3a, 4a...

xi. a, a^2, a^3, a^4…….

xii. √2,√8,√18,√32….

xiii. √3,√6,√9,√12…

xiv. 12, 32, 52, 72……

xv. 12, 52, 72, 73…..

Find the missing variable from a, d, n and an, where a is the first term, d is the common difference and an is the nth term of AP.

Choose the correct choice in the following and justify:

A. 30th term of the AP: 10, 7, 4,... is

i. 97

ii. 77

iii.–77

iv. –87

B. 11th term of the AP:−3, −12, 2,... is

i. 28

ii. 22

iii.–38

iv. - 481/2

In the following AP's find the missing terms:

(i) 2, ___, 26

(ii)  ___, 13, ___, 3

(iii) 5, ___, ___, 91/2

(iv) –4, ___, ___, ___, ___, 6

(v) ___, 38, ___, ___, ___, –22

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) 18, 15(1/2), 13, ….., −47

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 29th term.

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

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 is100, 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 is 44. Find the first three terms of the AP.

Subba Rao started work in 1995 at an annual salary of ₹ 5000 and received an increment of ₹ 200 each year. In which year did his income reach ₹ 7000?

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

Find the sum of the following AP's.

(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) 1/15, 1/12, 1/(10  ), …, to 11 terms.

Find the sums given below:

(i) 7 + 10(1/2) + 14 +……+ 84

(ii) 34 + 32 + 30 + … + 10

(iii) –5 + (–8) + (–11) + ... + (–230)

In an AP

(i) Given a = 5, d = 3, an = 50, find n and Sn.

(ii) Given a = 7, a13 = 35, find d and S13.

(iii) Given a12 = 37, d = 3, find a and S12.

(iv) Given a3 = 15, S10 = 125, find d and a10.

(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, Sn = 210, find n and d.

(viii) Given an = 4, d = 2, Sn = -14, find n and a.

(ix) Given a = 3, n = 8, S = 192, find d.

(ix) Given l = 28, S = 144, and there are total of 9 terms. Find a.

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 terms and the common difference.

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?

Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

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

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.

Show that a1, a2,……, an,…. form an AP where an is defined as below:

(i) an = 3 + 4n

(ii) an = 9 - 5n

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

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.

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 a certain date as follows: ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day, etc., the penalty for each succeeding day being

₹ 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?

A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If, each prize is ₹ 20 less than its preceding term, find the value of each of the prizes.

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 two trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center 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. (Take π = 22/7)

[Hint: Length of successive semicircles is l1, l2, l3, l4,.... 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 soon. In how many rows are the 200 logs placed and how many logs are in the top row?

In a potato race, a bucket is placed at the starting point, which is 5 meters from the first potato, and the other potatoes are placed 3 meters 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?

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

2 × 5 + 2 × (5 + 3)]

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

[Hint : Find n for an< 0]

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.

A ladder has rungs 25 cm apart (see figure). The rungs decrease uniformly in length from 45 cm, at the bottom to 25 cm at the top. If the top and the bottom rungs are 2(1/2) m apart, what is the length of the wood required for the rungs?

[Hint: Number of rungs = 250/25 + 1]

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.

[Hint: Sx – 1 = S49 – Sx]

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 at read of 1/2 m (see figure). Calculate the total volume of concrete required to build the terrace.

[Hint : Volume of concrete required to build the first step = 1/4 × 1/2 × 50 m^3]