Solution of Chapter 6. Progressions (AP Mathematics Book)

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

The taxi fare after each km when the fare is Rs. 20 for the first km and rises by Rs. 8 for each additional km.

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

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

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

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

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

The amount of money in the account every year, when Rs. 10000 is deposited at compound interest at 8% per annum.

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

a = 10, d = 10

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

a = -2, d = 0

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

a = 4, d = -3

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

a = 1, d = 1/2.

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

a = -1.25, d = -0.25

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

3, 1, -1,-3

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

-5, -1, 3, 7

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

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

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.

2, 4, 8, 16.

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

2, 5/2, 3, 7/2

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

-1.2, -3.2, -5.2, -7.2, . . .

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

-10, -6, -2, 2, . . .

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

3, 3 + √ 2, 3 + 2√ 2, 3 + 3√ 2 ……

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

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

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

0, -4, -8, -12, . . .

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

, , …..

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

1, 3, 9, 27, . . .

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

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

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

a, a², a³, a⁴

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

√2, √8, √18, √32

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

√3, √6, √9, √12

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

Find the

30^thterm of the A.P. 10, 7, 4 ……

Find the

11^thterm of the A.P. : -3, , 2, …

Find the respective terms for the following APs.

a₁ = 2, a₃ = 26, find a₂

Find the respective terms for the following APs.

a₂ = 13, a₄ = 3, find a₁, a₃

Find the respective terms for the following APs.

a₁ = 5, a₄ = 9, find a₂, a₃

Find the respective terms for the following APs.

a₁ = 4; a₆ = 6. Find a₂, a₃, a₄, a₅

Find the respective terms for the following APs.

a₂ = 38, a₆ = -22, find a₁, a₃, a₄, a₅

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

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

7, 13, 19, . . ., 205

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

18, 15, 13 …. , -47

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

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

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

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

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 7^thterm exceeds the 5^thterm by 12.

Find the 20^thterm from the end of the AP : 3, 8, 13, …., 253.

The sum of the 4th and 8^thterms of an AP is 24 and the sum of the 6^th and 10^th terms is 44. Find the first three terms of the AP.

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 salary reach Rs. 7000?

Find the sum of the following APs:

2, 7, 12, …., to 10 terms.

Find the sum of the following APs:

-37, -33, -29, ……, to 12 terms.

Find the sum of the following APs:

0.6, 1.7,2.8, ….., to 100 terms.

Find the sum of the following APs:

, , ….., to 11 terms

Find the sums given below:

7 + 10 + 14 +…. +84.

Find the sums given below:

34 + 32 + 30 +….+10

Find the sums given below:

(-5) + (-8) + (-11) +…… + (-230)

In an AP :

given a = 5, d = 3, a_n = 50 find n and s_n

In an AP :

given a = 7, a_n = 35 find d and s₁₃

In an AP :

given d = 3, a₁₂ = 37 find a and s₁₂

In an AP :

given S₁₀ = 125, a₃ = 15 find d and a₁₀

In an AP :

given a = 2, S_n = 90, d = 8 find n and a_n

In an AP :

given a_n = 4, d = 2, s_n = -14. Find n and a

In an AP :

given l = 28, s_n = 144 and there are 9 terms Find and a

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 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,…..a_n form an AP where is defined as below :

a_n = 3+4n

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

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

a_n = 9-5n

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

If the sum of first n terms of an AP is 4n-n² what is the first (remember the first term is s₁)? What is the sum of first two terms? What is the second term? Similarly, find the 3rd,and the 10th and the nth terms.

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

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.

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?

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 in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? Take π =

[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 so on. In how many rows are the 200 logs placed and how many logs are in the top row?

In a bucket and ball race, a bucket is placed at the starting point, which is 5 m from the first ball, and the other balls are placed 3 m apart in a straight line. There are ten balls in the line.

A competitor starts from the bucket, picks up the nearest ball, runs back with it, drops it in the bucket, runs back to pick up the next ball, runs to the bucket to drop it in, and she continues in the same way until all the balls are in the bucket. What is the total distance the competitor has to run?

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

In which of the following situations, does the list of numbers involved in the form of a GP.?

Salary of Sharmila, when her salary is Rs. 5,00,000 for the first year and expected to receive yearly increase of 10%.

In which of the following situations, does the list of numbers involved in the form of a GP.?

Number of bricks needed to make each step, if the stair case has total 30 steps. Bottom step needs 100 bricks and each successive step needs 2 brick less than the previous step.

In which of the following situations, does the list of numbers involved in the form of a GP.?

Perimeter of each triangle, when the mid points of sides of equilateral triangle whose side is 24 cm are joined to form another triangle, whose mid points in turn are joined to form still another triangle and the process continues indefinitely.

Write three terms of the GP when the first term ‘a’ and the common ratio ‘r’ are given?

a = 4, r = 3

Write three terms of the GP when the first term ‘a’ and the common ratio ‘r’ are given?

a = √5, r =

Write three terms of the GP when the first term ‘a’ and the common ratio ‘r’ are given?

a = 81, r =

Write three terms of the GP when the first term ‘a’ and the common ratio ‘r’ are given?

a = , r = 2

Which of the following are GP? If they are in GP. Write three more terms?

4, 8, 16,….

Which of the following are GP? If they are in GP. Write three more terms?

, ,……

Which of the following are GP? If they are in GP. Write three more terms?

5, 55, 555, …………

Which of the following are GP? If they are in GP. Write three more terms?

-2, -6, -18 ……..

Which of the following are GP? If they are in GP. Write three more terms?

……..

Which of the following are GP? If they are in GP. Write three more terms?

3, -3², 3³…….

Which of the following are GP? If they are in GP. Write three more terms?

x, 1, ……

Which of the following are GP? If they are in GP. Write three more terms?

Which of the following are GP? If they are in GP. Write three more terms?

0.4, 0.04, 0.004, ……

Find x so that x, x+2, x+6 are consecutive terms of a geometric progression.

For each geometric progression find the common ratio ‘r’, and then find a_n

3, ,,…..

For each geometric progression find the common ratio ‘r’, and then find a_n

2, -6, 18, -54….

For each geometric progression find the common ratio ‘r’, and then find a_n

-1, -3, -9, -27….

For each geometric progression find the common ratio ‘r’, and then find a_n

5, 2, …….

Find the 10^thand nth term of GP. : 5, 25, 125, …….

Find the indicated term of each geometric Progression

a₁ = 9, r = find a₇

Find the indicated term of each geometric Progression

a₁ = -12, r = find a₆

Which term of the GP.

2, 8, 32, ……. is 512?

Which term of the GP.

√3, 3, 3√3…… is 729?

Which term of the GP.

, ,,….is ?

Find the 12^th term of a GP. Whose 8^th term is 192 and the common ratio is 2.

The 4^th term of a geometric progression is and the seventh term is Find the geometric series.

If the geometric progressions 162, 54, 18 …. and have their nth term equal. Find the value of n.