Chapter 11 – Arithmetic Progressions Exercise Ex. 11A

Show that each of the progressions given

below is an AP. Find the first term, common difference and next term of each.

9, 15, 21, 27, …..

Show that each of the progressions given

below is an AP. Find the first term, common difference and next term of each.

11, 6, 1, –4, ….

Find

The 20^{th} term of the AP 9, 13,

17, 21, ….

Find

The 35^{th} term of the AP 20, 17,

14, 11, ….

Find

The 15^{th} term of the AP -40,

-15, 10, 35, ….

Find the 37^{th} term of the AP

The given AP is

First term = 6, common difference =

a = 6, d =

The n^{th} term is given by

Hence, 37^{th} term is 69

Find the 25^{th} term of the AP

The given AP is

The first term = 5,

common difference =

a = 5,

The n^{th} term is given by

Hence the 25^{th} term is – 7

Find the nth term of each of the following

APs :

5, 11, 17, 23, …..

Find the nth term of each of the following

APs :

16, 9, 2, –5, …..

If the nth term of a progression is (4n – 10), show that it is an A.P. Find its (i) first term, (ii) common difference, and (iii) 16^{th} term.

(i)First term = -6

(ii)Common difference

(iii)16^{th} term = where a = -6 and d = 4

= (-6 + 15 4) = 54

How many terms are there in the AP 6, 10, 14, 18, …, 174?

In the given AP, we have a = 6 and d = (10 – 6) = 4

Suppose there are n terms in the given AP, then

Hence there are 43 terms in the given AP

How many terms are there in the AP 41, 38, 35, …8?

In the given AP we have a = 41 and d = 38 – 41 = – 3

Suppose there are n terms in AP, then

Hence there are 12 terms in the given AP

Which term of the AP 3, 8, 13, 18, … is 88?

In the given AP, we have a = 3 and d = 8 – 3 = 5

Suppose there are n terms in given AP, then

Hence, the 18^{th} term of given AP is 88

Which term of the AP 72, 68, 64, 60, … is 0?

In the given AP, we have a = 72 and d = 68 – 72 = – 4

Suppose there are n terms in given AP, we have

Hence, the 19^{th} term in the given AP is 0

Which term of the AP is 3?

In the given AP, we have

Suppose there are n terms in given AP, we have

Then,

Thus, 14^{th} term in the given AP is 3

Which term of the AP 21, 18, 15, … is -81?

Which term of the AP 3, 8, 13, 18, … will be 55 more than its 20^{th} term?

The given AP is 3, 8, 13, 18…..

First term a = 3, common difference a = 8 – 3 = 5

Let n^{th} term is 55 more than the 20^{th} term

(5n – 2) – 98 = 55

Or 5n = 100 + 55 = 155

31^{st} term is 55 more than the 20^{th} term of given AP

Which term of the AP 5, 15, 25, … will be 130 more than its 31^{st} term?

The given AP is 5, 15, 25….

a = 5, d = 15 – 5 = 10

Thus, the required term is 44^{th}

If the 10^{th} term of an AP is 52 and the 17^{th} term is 20 more than the 13^{th} term, find the AP.

In the given AP let the first term = a,

And common difference = d

So the required AP is 7, 12, 17, 22….

Find the middle term of the AP 6, 13, 20, …, 216.

Find the middle term of the AP 10, 7, 4, …, (-62).

Find the 8^{th} term from the end of the AP 7, 10, 13, …, 184.

Here a = 7, d = (10 – 7) = 3, l = 184

And n = 8

Hence, the 8^{th} term from the end is 163

Find the 6^{th} term from the end of the AP 17, 14, 11, …(-40).

Here a = 17, d = (14 – 17) = -3, l = -40

And n = 6

Now, n^{th} term from the end = [l – (n – 1)d]

Hence, the 6^{th} term from the end is – 25

Is 184 a term of the AP 3, 7, 11, 15, …?

Is-150 a term of the AP 11, 8, 5, 2, …?

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

The 7^{th} term of an AP is -4 and its 13^{th} term is -16. Find the AP.

In the given AP, let the first term = a common difference = d

So the required AP is 8, 6, 4, 2, 0……

The 4^{th} term of an AP is zero. Prove that its 25^{th} term is triple its 11^{th} term.

Let the first term of given AP = a and common difference = d

Hence 25^{th} term is triple its 11^{th} term

The 8^{th} term of an AP is zero.

Prove that its 38^{th} term is triple its 18^{th} term.

The 4^{th} term of an AP is 11. The

sum of the 5^{th} and 7^{th} terms of this AP is 34. Find its

common difference.

The 9^{th} term of an AP is -32 and

the sum of its 11^{th} and 13^{th} terms is -94. Find the

common difference of the AP.

Determine the nth term of the AP whose 7th

term is -1 and 16th term is 17.

If 4 times the 4th term of an AP is equal

to 18 times its 18th term then find its 22nd term.

If 10 times the 10^{th} term of an AP is equal to 15 times the 15^{th} term, show that its 25^{th} term is zero.

Let a be the first term and d be the common difference

Find the common difference of an AP whose

first term is 5 and the sum of its first four terms is half the sum of the

next four terms.

The sum of the 2nd and the 7th terms of an

AP is 30. If its 15th term is 1 less than twice its

8th term, find the AP.

For what value of n, the n^{th} terms of the APs 63, 65, 67, … and 3, 10, 17, … are equal?

First AP is 63, 65, 67….

First term = 63, common difference = 65 – 63 = 2

n^{th}term = 63 + (n – 1) 2 = 63 + 2n – 2 = 2n + 61

Second AP is 3, 10, 17 ….

First term = 3, common difference = 10 – 3 = 7

n^{th} term = 3 + (n – 1) 7 = 3 + 7n – 7 = 7n – 4

The two n^{th} terms are equal

2n + 61 = 7n – 4 or 5n = 61 + 4 = 65

The 17th term of AP is 5 more than twice

its 8th term. If the 11th term of the AP is 43, find its nth term.

The 24th term of an AP is twice its 10th

term. Show that its 72nd term is 4 times its 15th term.

The 19th term of an AP is equal to 3 times

its 6th term. If its 9th term is 19, find the AP.

If the p^{th} term of an AP is q and its q^{th} term is p, then show that its (p + q)^{th} term is zero.

Let a be the first term and d be the common difference

p^{th} term = a +(p – 1)d = q(given)—–(1)

q^{th} term = a +(q – 1) d = p(given)—–(2)

subtracting (2) from (1)

(p – q)d = q – p

(p – q)d = -(p – q)

d = -1

Putting d = -1 in (1)

The first and last terms of an AP are a and l respectively. Show that the sum of the n^{th} term from the beginning and the n^{th} term from the end is (a + l).

Let a be the first term and d be the common difference

n^{th} term from the beginning = a + (n – 1)d—–(1)

n^{th} term from end= l – (n – 1)d —-(2)

adding (1) and (2),

sum of the n^{th} term from the beginning and n^{th} term from the end = [a + (n – 1)d] + [l – (n – 1)d] = a + l

How many two-digit numbers are divisible by

6?

How many two-digit numbers are divisible by

3?

How many three-digit numbers are divisible

by 9?

How many numbers are there between 101 and

999, which are divisible by both 2 and 5?

In a flower bed, there are 43 rose plants in the first row, 41 in the second, 39 in the third, and so on. There are 11 rose plants in the last row. How many rows are there in the flower bed?

Number of rose plants in first, second, third rows…. are 43, 41, 39… respectively.

There are 11 rose plants in the last row

So, it is an AP . viz. 43, 41, 39 …. 11

a = 43, d = 41 – 43 = -2, l = 11

Let n^{th} term be the last term

Hence, there are 17 rows in the flower bed.

A sum of Rs. 2800 is to be used to award

four prizes. If each prize after the first is Rs. 200 less than the preceding prize, find

the value of each of the prizes.

## Chapter 11 – Arithmetic Progressions Exercise Ex. 11B

Determine

k so that (3k – 2), (4k – 6) and (k + 2) are three consecutive terms of an

AP.

Find

the value of x for which the numbers (5x + 2),(4x – 1) and (x + 2) are in AP.

If

(3y – 1),(3y + 5) and (5y + 1) are three consecutive terms of an AP then find

the value of y.

Find the value of x for which (x + 2), 2x, (2x + 3) are three consecutive terms of an AP.

If are consecutive terms of an AP, then

Find three numbers in AP whose sum is 15 and product is 80.

Let the required numbers be (a – d), a and (a + d)

Sum of these numbers = (a – d) + a + (a + d) = 3a

Product of these numbers = (a – d) × a × (a + d) =

But sum = 15 and product = 80

Hence, the required numbers are (2, 5, 8)

The sum of three numbers in AP is 3 and their product is -35. Find the numbers.

Let the required numbers be (a – d), a, (a + d)

Sum of these number = (a – d) + a + (a + d) = 3a

Product of these numbers = (a – d) a (a + d)

But,sum = 3 and product = – 35

Thus, a = 1 and d = 6

Hence, the required numbers are (-5, 1, 7)

Divide 24 into three parts such that they are in AP and their product is 440.

Let the required number be (a – d), a and (a + d)

Sum of these numbers = (a – d) + a + (a + d) = 3a

Product of these numbers = (a – d) x a x (a + d)

But sum = 24 and product = 440

Thus, a = 8 and d = 3

Hence the required numbers are (5, 8, 11)

The sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. Find these terms.

Let the required numbers be (a – d), a, (a + d)

Sum of these numbers = (a – d) + a + (a + d) = 3a

sum of these squares =

Sum of three numbers = 21, sum of squares of these numbers = 165

3a = 21

a = 7

Thus, a = 7 and d =

Hence, the required numbers are (4, 7, 10) or (10, 7, 4)

The angles of a quadrilateral are in AP whose common difference is 10°. Find the angles.

Let the required angles be (a – 3d)°, (a – d) °, (a + d) ° and (a + 3d) °

Common difference = (a – d) – (a- 3d) = a – d – a + 3d = 2d

Common difference = 10°

2d = 10° = d = 5°

Sum of four angles of quadrilateral = 360°

First angle = (a – 3d)° = (90 – 3 × 5) ° = 75°

Second angle = (a – d)° = (90 – 5) ° = 85°

Third angle = (a + d)° = (90 + 5°) = 95°

Fourth angle = (a + 3d)° = (90 + 3 × 5)° = 105°

Find four numbers in AP whose sum is 28 and the sum of whose squares is 216.

Let the required number be (a – 3d), (a – d), (a + d) and (a + 3d)

Sum of these numbers = (a – 3d) + (a – d)+ (a + d) + (a + 3d)

4a = 28 a = 7

Sum of the squares of these numbers

Hence, the required numbers (4, 6, 8, 10)

Divide

32 into four parts which are the four terms of an AP such that the product of

the first and the fourth terms is to the product of the second and the third

terms as 7 : 15.

The

sum of first three terms of an AP is 48. If the product of first and second

terms exceeds 4 times the third term by 12. Find the AP.

## Chapter 11 – Arithmetic Progressions Exercise Ex. 11C

The first three terms of an AP are

respectively (3y – 1), (3y + 5) and (5y + 1), find the value of y.

If k, (2k – 1) and (2k + 1) are the three

successive terms of an AP, find the value of k.

If 18, a, (b – 3) are in AP, then find the

value of (2a – b).

If the numbers a, 9, b, 25 form an AP, find

a and b.

If the numbers (2n – 1), (3n + 2) and (6n –

1) are in AP, find the value of n and the numbers.

How many three-digit natural numbers are

divisible by 7?

How many three-digit natural numbers are

divisible by 9?

If the sum of first m terms of an AP is (2m^{2}

+ 3m) then what is its second term?

What is the sum of first n terms of the AP

a, 3a, 5a, ….

What is the 5th term from the end of the AP

2, 7, 12, …, 47?

If a_{n} denotes the nth term of

the AP 2, 7, 12, 17, …, find the value of (a_{30}

– a_{20}).

Find the sum of all

three-digit natural numbers which are divisible by 13.

All three digit

natural numbers divisible by 13 are 104, 117, 130, 143,…, 988

This is an AP in

which a = 104, d = (117 – 104) = 13, l = 988

Find the sum of first 15 multiples of 8.

First 15 multiples of 8 are 8, 16, 24, … to 15^{th} term

Find the sum of all odd numbers between 0 and 50.

Odd natural numbers between 0 and 50 are 1, 3, 5, … 49

a = 1, d = 3 – 1= 2, l = 49

Let the number of terms be n

Find the sum of first hundred even natural numbers which are divisible by 5.

First 100 even natural numbers divisible by 5 are

10, 20, 30, … to 100 term

First term of AP = 10

Common difference d = 20 – 10 = 10

Number of terms = n = 100

Which term of the AP 21, 18, 15, … is zero?

The given AP is 21, 18, 15, ….

First term = 21, common difference = 18 – 21= – 3

Let n^{th} term be zero

a + (n – 1)d = 0or 21 + (n – 1)(-3) = 0

21 – 3n + 3 = 0

3n = 24

or

Hence, 8^{th} term of given series is 0

Find the sum of first n natural numbers.

Sum of n natural numbers = 1 + 2 + 3 + … + n

Here a = 1, d = 2 – 1 = 1

Find the sum of first n even natural numbers.

Sum of even natural numbers = 2 + 4 + 6 + … to n terms

a = 2, d = 4 – 2 = 2

The first term of an AP is p and its common difference is q. Find its 10^{th} term.

First term of AP = a = p

Common difference = d = q

n^{th} term = a + (n 1)d

10^{th} term = p + (10 1)q

= p + 9q

If are three consecutive terms of an AP, find the value of a.

If (2p + 1), 13, (5p – 3) are in AP, find

the value of p.

If (2p – 1), 7, 3p are in AP, find the

value of p.

If the sum of first p terms of an AP is (ap^{2 }+ bp), find its common difference.

If the sum of first n terms is (3n^{2}

+ 5n), find its common difference.

Find an AP whose 4th term is 9 and the sum

of its 6th and 13th terms is 40.

## Chapter 11 – Arithmetic Progressions Exercise Ex. 11D

Find

the sum of each of the following APs:

2,

7, 12, 17, ….. to 19 terms.

Find

the sum of each of the following APs:

9,

7, 5, 3, ….. to 14 terms.

Find

the sum of each of the following APs:

-37,

-33, –29, …. to 12 terms.

Find

the sum of each of the following APs:

Find

the sum of each of the following APs:

0.6,

1.7, 2.8, ….. to 100 terms.

Find

the sum of each of the following arithmetic series:

Find

the sum of each of the following arithmetic series:

Find

the sum of each of the following arithmetic series:

Find

the sum of first n terms of an AP whose n^{th} term is (5 – 6n). Hence,

find the sum of its first 20 terms.

The sum of n terms of an AP is . Find its 20^{th} term.

It is given that —–(1)

Now, 20^{th} term

=(sum of first 20 term) – (sum of first 19 terms)

Putting = 20 in (1) we get

Putting n= 19 in (1), we get

Hence, the 20^{th}term is 99

If the sum of the first n terms of an AP is given by S_{n} = (3n^{2} – n), find its (i) nth term, (ii) first term and (iii) common difference.

(i)The nth term is given by

(ii)Putting n = 1 in (1) , we get

(iii)Putting n = 2 in (1), we get = 8

How many terms of the AP 21, 18, 15, … must be added to get the sum 0?

Here a = 21, d = (18 – 21) = -3

Let the required number of terms be n, then

sum of first 15 terms = 0

How

many terms of the AP 9, 17, 25, …. must be taken so that their sum is 636?

How

many terms of the AP 63, 60, 57, 54, … must be taken

so that their sum is 693? Explain the double answer.

Write the next term of the AP

The term AP is

Find

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

Find

the sum of first forty positive integers divisible by 6.

The nth term of an AP is (7 – 4n). Find its common difference.

Find the sum of all multiples of 9 lying between 300 and 700.

All numbers between 300 and 700 that are multiples of 9 are 306, 315, 324, 333, …, 693

This is an AP in which a = 306, d = (315 – 306) = 9, l = 693

Let the number of these terms be n, then

The nth term of an AP is (3n + 5). Find its common difference.

Thus, common difference = 3

Write the next term of the AP

The given AP is

Common difference d =

Term next to

Find the sum of the following:

The given AP is

First term

Common difference d =

Sum of n terms =

In an AP the first term is 2, the last term is 29 and sum of the terms is 155. Find the common difference of the AP.

First term ‘a’ of an AP = 2

The last term l = 29

common difference = 3

In

an AP, the first term is -4, the last term is 29 and the sum of all its terms

is 150. Find its 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?

The

first and the last terms of an AP are 5 and 45 respectively. If the sum of

all its terms is 400, find the common difference and the number of terms.

In an AP the first term is 22, n^{th} term is -11 and sum to first n^{th} terms is 66. Find n and d, the common difference.

First term of an AP, a = 22

Last term = n^{th} term = – 11

Thus, n = 12, d = -3

The

12^{th} term of an AP is -13 and the sum of its first four terms is

24. Find the sum of its first 10 terms.

The

sum of the first 7 terms of an AP is 182. If its 4^{th} and 17^{th}

terms are in the ratio 1 : 5, find the AP.

The

sum of the first 9 terms of an AP is 81 and that of its first 20 terms is

400. Find the first term and the common difference of the AP.

The

sum of the first 7 terms of an AP is 49 and the sum of its first 17 terms is

289. Find the sum of its first n terms.

Two

APs have the same common difference. If the first terms of these APs be 3 and

8 respectively, find the difference between the sums of their first 50 terms.

The

sum of first 10 terms of an AP is -150 and the sum of its next 10 terms is

-550. Find the AP.

The

13^{th} term of an AP is 4 times its 3^{rd} term. If its 5^{th}

term is 16, find the sum of its first 10 terms.

The

16^{th} term of an AP is 5 times its 3^{rd} term. If its 10^{th}

term is 41, find the sum of its first 15 terms.

An

AP 5, 12, 19, … has 50 terms. Find its last term.

Hence, find the sum of its last 15 terms.

An

AP 8, 10, 12, … has 60 terms. Find its last term.

Hence, find the sum of its last 10 terms.

The

sum of the 4^{th} and the 8^{th} terms of an AP is 24 and the sum of its 6^{th} and 10^{th}

terms is 44. Find the sum of its first 10 terms.

The

sum of first m terms of an AP is (4m^{2} – m).If its nth term is 107,

find the value of n. Also ,find the 21^{st} term of this AP.

The

sum of first q terms of an AP is (63q -3q^{2}). If its pth term is -60, find the value of p. Also , find the 11^{th} term of its AP.

Find

the number of terms of the AP -12, -9, -6, …, 21.

If 1 is added to each term of this AP then find the sum of all terms of the

AP thus obtained.

Sum

of the first 14 terms of an AP is 1505 and its first term is 10. Find its 25^{th}

term.

Find

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

respectively.

In

a school, students decided to plant 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 double of the class in which they are studying.

If there are 1 to 12 classes in the school and each class has two sections,

find how many trees were planted by students. Which value is shown in the

question?

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 10 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 he continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

There are 25 trees at equal distances of 5 m in a line with a water tank, the distance of the water tank from the nearest tree being 10 m. A gardener waters all the trees separately, starting from the water tank and returning back to the water tank after watering each tree to get water for the next. Find the total distance covered by the gardener in order to water all the trees.

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 prize.

A

man saved Rs. 33000 in 10 months. In each month after the first, he saved Rs.

100 more than he did in the preceding month. How much did he save in the

first month?

A

man arranges to pay off a debt of Rs. 36000 by 40 monthly installments which

form an arithmetic series. When 30 of the installments are paid, he dies

leaving one-third of the debt unpaid. Find the value of the first installment.

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 1 contractor has to pay as penalty, if he

has delayed the work by 30 days?

## Chapter 11 – Arithmetic Progressions Exercise MCQ

If 4, x_{1}, x_{2}, x_{3}, 28 are in AP then x_{3}= ?

(a) 19

(b) 23

(c) 22

(d) cannot be determined

If the nth term of an AP is (2n + 1) then the sum of its first three terms is

(a) 6n + 3

(b) 15

(c) 12

(d) 21

The sum of first n terms of an AP is (3n^{2} + 6n). The common difference of the AP is

(a) 6

(b) 9

(c) 15

(d) -3

The sum of first n terms of an AP is (5n – n^{2}). The nth term of the AP is

(a) (5 – 2n)

(b) (6 – 2n)

(c) (2n – 5)

(d) (2n – 6)

The sum of first n terms of an AP is (4n^{2} + 2n). The nth term of this AP is

(a) (6n – 2)

(b) (7n – 3)

(c) (8n – 2)

(d) (8n + 2)

The 7th term of an AP is -1 and its 16th term is 17. The nth term of AP is

(a) (3n + 8)

(b) (4n – 7)

(c) (15 – 2n)

(d) (2n -15)

The 5th term of an AP is -3 and its common difference is -4. The sum of its first 10 terms is

(a) 50

(b) -50

(c) 30

(d) -30

The 5th term of an AP is 20 and the sum of its 7th and 11th terms is 64. The common difference of the AP is

(a) 4

(b) 5

(c) 3

(d) 2

The 13th term of an AP is 4 times its 3rd term. If its 5th term is 16 then the sum of its first ten terms is

(a) 150

(b) 175

(c) 160

(d) 135

An AP 5, 12, 19, … has 50 terms. Its last term is

(a) 343

(b) 353

(c) 348

(d) 362

The sum of first 20 odd natural numbers is

(a) 100

(b) 210

(c) 400

(d) 420

The sum of first 40 positive integers divisible by 6 is

(a) 2460

(b) 3640

(c) 4920

(d) 4860

How many two-digit numbers are divisible by 3?

(a) 25

(b) 30

(c) 32

(d) 36

How many three-digit numbers are divisible by 9?

(a) 86

(b) 90

(c) 96

(d) 100

What is the common difference of an AP in which a_{18} – a_{14} = 32?

(a) 8

(b) -8

(c) 4

(d) -4

If a_{n} denotes the nth term of the AP 3, 8, 13, 18, … then what is the value of (a_{30} -a_{20})?

(a) 40

(b) 36

(c) 50

(d) 56

Which term of the AP 72, 63, 54, … is 0?

(a) 8^{th}

(b) 9^{th}

(c) 10^{th}

(d) 11^{th}

Which term of the AP 25, 20, 15, … is the first negative term?

(a) 10^{th}

(b) 9^{th}

(c) 8^{th}

(d) 7^{th}

Which term of the AP 21, 42, 63, 84, … is 210?

(a) 9^{th}

(b) 10^{th}

(c) 11^{th}

(d) 12^{th}

What is 20th term from the end of the AP 3, 8, 13, …, 253?

(a) 163

(b) 158

(c) 153

(d) 148

(5 + 13 +21+… + 181) =?

(a) 2476

(b) 2337

(c) 2219

(d) 2139

The sum of first 16 terms of the AP 10, 6, 2, … is

(a) 320

(b) -320

(c) -352

(d) -400

How many terms of the AP 3, 7, 11, 15, … will make the sum 406?

(a) 10

(b) 12

(c) 14

(d) 20

The 2^{nd} term of an AP is 13 and its 5^{th} term is 25. What is its 17^{th} term?

(a) 69

(b) 73

(c) 77

(d) 81

The 17^{th} term of an AP exceeds its 10^{th} term by 21. The common difference of the AP is

(a) 3

(b) 2

(c) -3

(d) -2

The 8^{th} term of an AP is 17 and its 14^{th} term is 29. The common difference of the AP is

(a) 3

(b) 2

(c) 5

(d) -2

The 7^{th} term of an AP is 4 and its common difference is -4. What is its first term?

(a) 16

(b) 20

(c) 24

(d) 28