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SSLC Maths Chapter 1 Arithmetic Sequences Questions and Answers
Arithmetic Sequences Class 10 Questions and Answers Kerala State Syllabus
SCERT Class 10 Maths Chapter 1 Arithmetic Sequences Solutions
Class 10 Maths Chapter 1 Kerala Syllabus – Number Sequence
Textbook Page No. 8, 9
Question 1.
We can make triangles by stacking dots:
Write the number of dots in each triangle. Calculate the number of dots needed to make the next three triangles in this pattern.
Answer:
a) 3, 6, 10…
b) 15, 21
Question 2.
From the sequence equilateral triangle, square, regular pentagon and so on of regular polygons form the following sequences .
a) Number of sides 3, 4, 5, 6
b) Sum of inner angles
c) Sum of outer angles
d) An inner angle
e) An outer angle
Answer:
a) 3, 4, 5, 6……….
b) 180°, 360°, 540°…
c) 360°, 360°, 360°, 360°…
d) 60°, 90°, 108°, 120°…
e) 120°, 90°, 72°, 60°…
Question 3.
Write the sequence of natural numbers which leave remainder 1 on division by 3 , and the sequence of natural numbers which leave remainder 2 on division by 3
Answer:
a) 1, 4, 7, 10…
b) 2, 5, 8, 11…
Question 4.
Write in ascending order, the sequence of natural numbers with last digit 1 or 6. Describe this sequence in two other ways
Answer:
1,6, 11,16…
The sequence of numbers gives 1 as remainder on dividing by 5
Sequence of numbers 4 less than multiples of 5
Question 5.
See these figures :
The first picture shows an equilateral triangle with the smaller triangle got by joining the midpoints of sides cut off. The second picture shows the same thing done on each of the three triangles in the first picture. The third picture shows the same thing done on the second picture
(i) How many red triangles are there in each picture?
Answer:
3, 9, 27, ……………………..
(ii) Taking the area of whole uncut triangle as 1, compute the area of a small triangle in each picture.
Answer:
\(\frac{1}{4}, \frac{1}{16}, \frac{1}{64}\)……
(iii) What is the total area of all the red triangles in each picture?
Answer:
\(\frac{3}{4}, \frac{9}{16}, \frac{27}{64}\) ………….
(iv) Write the first five terms of each of the three sequences got by continuing this process
Answer:
3, 9, 27, 81, 243 ……..
\(\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1024}\) ……….
\(\frac{3}{4}, \frac{9}{16}, \frac{27}{64}, \frac{81}{256}, \frac{253}{1024}\) ………….
Class 10 Maths Kerala Syllabus Chapter 1 Solutions – Arithmetic Sequence
Textbook Page No. 13
Question 1.
Check whether each of the sequences given below are arithmetic sequences. Give reasons also. Find the common differences of the arithmetic sequences:
a) Natural numbers leaving remainder 1 on division by 4
Answer:
1, 5, 9 ….
Here the sequence starting from 1 and adding 4 again and again.
So it is an arithmetic sequence.
Therefore the common difference is 4
b) Natural numbers leaving remainder 1 or 2 on division by 4
Answer:
1,2, 5, 6, 9, 10 …………….
Here the common difference does not exists.
Therefore it is not an arithmetic sequence.
c) Square of natural numbers
Answer:
1, 4, 9, 16, ….
Here the common difference does not exists.
Therefore it is not an arithmetic sequence.
d) Reciprocals of natural numbers
Answer:
1, \(\frac{1}{2}\), \(\frac{1}{3}\), ……….
Here the common difference does not exists.
Therefore it is not an arithmetic sequence,
e) Powers of 2
Answer:
2, 4, 8, ….
Here the common difference does not exists.
Therefore it is not an arithmetic sequence.
f) Half of the odd numbers
Answer:
\(\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\)
Here the sequence starting from \(\frac{1}{2}\) and adding 1 again and again.
So it is an arithmetic sequence.
Therefore the common difference is 1
Question 2.
See these pictures:
a) How many small squares are there in each picture
b) How many large squares?
c) How many squares in all in each picture?
If we continue the pattern of pictures, are the se- 3. quences above arithmetic sequences?
Answer:
a) 2, 4, 6, 8,….
Here the sequence starting from 2 and adding 2 again and again.
Therefore it is an arithmetic sequence.
b) 0, 1,2,3, ….
Here the sequence starting from 0 and adding 1 again and again.
Therefore it is an arithmetic sequence.
c) 2,5,8, 11,….
Here the sequence starting from 0 and adding 1 again and again.
Therefore it is an arithmetic sequence. Whenever this pattern continues like this, the above sequences are also an arithmetic sequence.
Question 3.
In the picture below, the perpendiculars drawn from the bottom line are equally spaced.
Show that the sequence of the heights of the perpendiculars, on continuing this, form an arithmetic sequence.
(Hint: Draw perpendiculars from the top of each perpendicular to the next perpendicular)
Answer:
Lengths of vertical lines x, x +y, x + 2y, x + 3y, x + 4y
y is the difference between the term which is common everywhere
These are in arithmetic sequence.This is an arith-metic sequence having 5 terms.
SCERT Class 10 Maths Chapter 1 Solutions – Position and Term
Textbook Page No. 15
Question 1.
In each of the arithmetic sequences below, some of the terms are not written, but indicated by ⬭. Find out these numbers
a) 24, 42, ⬭, ⬭, …..
b) ⬭, 24, 42, ⬭ ……….
c) ⬭, ⬭, 24, 42 ……..
d) 24, ⬭, 42, ⬭, ……….
e) ⬭, 24, ⬭, 42 ……..
f) 24, ⬭ , ⬭, 42 ………
Answer:
a) Common difference is 42 – 24 = 18
Third term is 42 + 18 = 60,
Fourth term is 60 + 18 = 78
b) Common difference is 42 – 24 = 18
First term is 24 – 18 = 6,
Fourth term is 42 + 18 = 60
c) Common difference is 42 – 24 = 18
Second term 24 – 18 = 6
First term is 6 – 18 = -12
d) Difference 42 – 24 = 18 is two times the common difference.
Common difference is 9, second term is 24 + 9 = 33, fourth term is 42 + 9 = 51
e) Difference between 24 and 42 is 18. It is two times common difference.
Common difference is 9.
First term is 24 – 9 = 15, third term is 33
f) Difference between 42 and 24 is 18
Three times common difference is 6.
Second term is 30, Third term 36.
Question 2.
The two terms in specific positions of some arith-metic sequences are given below. Write first five terms of each.
(a) 3rd term 34 and 6th term 67
(b) 3rd term 43 and 6th term 76
(c) 3rd term 2 and 5th term 3
(d) 5th term 8 and 9th term 10
(e) 5th term 7 and 7th term 5
Answer:
(a) Difference between third term and sixth term is 3 times common difference.
3 × common difference = 67 – 34 = 33, common difference is 11.
Therefore the sequence is 12, 23, 34, 45, 56, 67.
(b) 3 times common difference is 76 – 43 = 33.
The common difference is 11 First term = third term – two times common difference
3rd term = 43 – 2 11 = 21 Terms are 21, 32, 43, 54, 65
(c) 2 times common difference is 3 – 2 = 1
Common difference is \(\frac{1}{2}\)
First term is = 2 – 2 × \(\frac{1}{2}\) =1 2
Terms are 1, 1 \(\frac{1}{2}\) ,2, 2 \(\frac{1}{2}\), 3
(d) 4 times common difference is 10 – 8 = 2.
Common difference is \(\frac{1}{2}\)
First term = 5 th term – 4 × \(\frac{1}{2}\)
Terms are 6, 6 \(\frac{1}{2}\), 7, 7\(\frac{1}{2}\), 8
(e) 2 times the common difference is 5 – 7 = -2
Common difference = -1
First term = 5th term – 4 × (-1) = 11
Terms are = 11, 10, 98, 7
SSLC Maths Chapter 1 Questions and Answers – Change In Position And Terms
Textbook Page No. 18
Question 1.
What is the 25th term of the arithmetic sequence 1, 11, 21…?
Answer:
25th term = 1st term + 24 times common difference 25 th term is 1 + 24
10 = 241
Question 2.
10th term of an arithmetic sequence is 46 and 11th term is 51
a) What is the first term?
b) Write the first five terms of the sequence.
Answer:
a) Common difference is 5
First term = 10th term – 9 times common difference
First term = 46 – 9 × 5 = 1
b) 1, 6, 11, 16, 21
Question 3.
What is the 21st term of the arithmetic sequence 100, 95, 90 ….
Answer:
Common difference is -5
21st term = First term + 20 times common difference
21st term = 100 + 20 × -5 = 0
Question 4.
The 10th term of an arithmetic sequence is 56 and its 11th term is 51,
a) What is the first term?
b) Write the first five terms of the sequence.
Answer:
Common difference is -5
a) First term = 56 – 9 × -5 = 56 + 45 = 101
b) 101, 96, 91, 86, 81
Textbook Page No. 21
Question 1.
The 3rd term of an arithmetic sequence is 15 and the 8th term is 35
(i) What is its 13th term?
(ii) What is its 23rd term?
Answer:
(i) 5 times the common difference = 35 – 15 = 20
13th term = 8th term + 5 times common difference = 35 + 20 = 55
(ii) 23rd term = 13th term + 10 times common difference = 55 + 40 = 95
2. The 5th term of an arithmetic sequence is 21 and the 9th term is 41
(i) What is its first term?
(ii) What is its 3rd term?
Answer:
(i) 4 times the common difference = 41 – 21 = 20 1st term = 5th term – 4 times common difference
= 21 – 20 = 1
(ii) 3rd term = 151 term + 2 times common difference = 1 + 10 = 11
Question 3.
The 4th term of an arithmetic sequence is 61 and the 7th term is 31
(i) What is its 10th term?
(ii) What is its first term?
Answer:
(i) Three common difference is -30
10th term = 4th term + 6 times common difference.
10th term = 61 + (-60) = 1
(ii) First term = 4th term – 3 times common difference
First term = 61 – (-30) = 91
Question 4.
The 5th term of an arithmetic sequence is 10 and the 10th term is 5
(i) What is its 15th term?
(ii) What is its 25th term?
Answer:
(i) Fifth term of an arithmetic sequence is 10 and tenth term is 5. Find 15th term and 25th term.
5 times common difference is 5- 10 = -5
15th term = 10th term + 5 times common difference
Arithmetic Sequences
15th term = 5 + (-5) = 0
(ii) 25th term =15th term + ten times common difference
25th term = 0 + -10 = -10
Textbook Page No. 22
Question 1.
Is 101 a term of the arithmetic sequence 13, 24, 35.. . ? What about 1001?
Answer:
We know that difference between two terms of an arithmetic sequence is divisible by the common difference.
101 – 13 = 88; is divisible by 11.
101 is a term of the sequence.
1001 – 13 = 988. It is not divisible by 11.
1001 is not a term
Question 2.
In the table below, some arithmetic sequences are given and two numbers against each. Check whether the numbers are terms of the respective sequences:
Answer:
Verification
In the first sequence, 123 – 11 = 112. It is not divisible by 11.
Therefore 123 is not a term 132 – 11 = 121. It is divisible by 11.
Therefore 132 is a term.
In the second sequence, 100 -; 12 = 88 , is divisible by 11
100 is a term
1000 – 21= 988 is not divisible by 11 .Therefore 1000 is not a term
100 – 1 = 79 is not divisible by 11 Therefore 100 is not a term
1000 – 21 = 979 is divsible by 11 .Therefore 1000 is a term
3 – \(\frac{1}{4}\) is 2\(\frac{3}{4}\). It is divisible by \(\frac{1}{4}\) 3 is a term
4 is also a term
In the next case, 3 is a term. 4 is not a term
Question 3.
In the table above find the position of numbers that are terms of the respective sequences.
Answer:
In the first sequence 132 = 11 + 11 × 11. That is, 132 is the 12th term
In the second sequence 100 = 12 + 8 × 11. 100 is 9th term.
In the third sequence 1000 = 21 + 89 × 11. It is 90th term.
3 = \(\frac{1}{4}\) + 11 × \(\frac{1}{4}\)
3 is 12th term
4 = \(\frac{1}{4}\) + 15 × \(\frac{1}{4}\)
It is 16th term.
3 is a term. 3 = \(\frac{3}{4}\) + 3 × \(\frac{3}{4}\)
It is 4th term.
Textbook Page No. 26
Question 1.
4th term of an arithmetic sequence is 8.
a) What is the sum of 3rd and 5th terms?
b) What is the sum of 2nd and 6th terms?
c) What is the sum of 1st and 7th terms?
Answer:
a) 2 × 8 = 16
b) 2 × 8 = 16
c) 2 × 8 = 16
Question 2.
4th term of an arithmetic sequence is 8
a) What is the sum of 3rd, 4th and 5th terms of this sequence?
b) What is the sum of the terms from 2nd to 6th ?
c) What is the sum of terms from 1st to 7th.
Answer:
a) 8 × 3 = 24
b) 8 × 5 = 40
c) 8 × 7 = 56
Question 3.
The common difference of an arithmetic sequence is 2 and sum of the 9th, 10th and 11th terms is 90. Calculate the first three terms of the sequence.
Answer:
Tenth term = \(\frac{90}{3}\) = 30
First term = tenth term – 9 × common difference = = 30 – 18 = 12
12, 14, 16 ….
Textbook Page No. 27
Question 1.
Write three arithmetic sequences with the sum of first 7 terms is 70
Answer:
4th term = \(\frac{70}{7}\) = 10
Arithmetic sequences can be formed by choosing different common differences.
If common difference is 1, sequence: 7, 8, 9, 10, 11, 12, 13 ………
If common difference is 2, sequence: 4, 6, 8, 10, 12, 14, 16…………
If common difference is 3, sequence: 1, 4, 7, 10, 13, 16, 19……….
Question 2.
The sum of the first 3 terms of an arithmetic sequence is 30, sum of the first 7 terms is 140
a) What is the 2nd term of the sequence?
b) What is the 4th term of the sequence?
c) What are the first three terms of the sequence?
Answer:
a) 2nd term = \(\frac{30}{3}\) = 10
b) 4th term is \(\frac{140}{7}\) = 20
c) First three terms = 5, 10, 15
Question 3.
The Sum of the first 5 terms of an arithmetic sequence is 150 and the sum of the first 10 terms is 550
a) What is the third term?
b) What is the 8th term?
c) What are the first five terms of the sequence?
Answer:
a) Third term = \(\frac{150}{5}\) = 30
b) 3rd term + 8th term = 110
8th term = 110 – 30 = 80
c) 5 times common difference is 80 – 30 = 50
Common difference = \(\frac{50}{5}\) = 10
sequence is 10, 20, 30…
Question 4.
Sum of the 11th and 21st terms of an arithmetic sequence is 80. What is 16th term?
Answer:
There are 11 term from 11th term to 21 term.
Since number of terms is odd there will a middle term
16th term is the middle term
16th term is half of the sum of 11th term and 21st term.
16th term is 40.
Question 5.
Angles of a pentagon are in arithmetic sequence.
a) If the angles are written in according to their magnitude, what would be the third angle?
b) If the smallest angle is 40° then what are the other angles?
c) Can the smallest angle be 36°?
Answer:
a) Angle sum of the polygon is (n – 2) 180 = 540°
Third angle \(\frac{540}{5}\) = 108°
b) 3rd term – 1st term = 2 × common difference 2 x common difference = 108 – 40 = 68.
Common difference = \(\frac{68}{2}\) = 34
Angles are 40°, 74°, 108°, 142°, 176°
c) Assume 36° as the smallest angle. The middle angle or third term is 108.
Difference between 3rd term and 1st term is 108 – 36 = 72
Fifth term will be 108 + 72 = 180.
Angle of a polygon cannot be 180°. That is smallest angle should be greater than 36°.
Textbook Page No. 30
Question 1.
Write 4 arithmetic sequences with sum of first 4 terms is 100
Answer:
Sum of first term and fourth term is 50. Sum of second and third terms is 50
a) Sequence 1.
If we choose second term as 10 then third term will be 40
Common difference is 40 – 10 = 30
First term 10 – 30 = – 20
– 20, 10, 40, 70
b) Sequence 2
If we choose second term as 20 then third term will be 30
Common difference is 30 – 20 = 10
First term 20 – 10 = 10
10, 20, 30, 40
c) Sequence 3
If we choose second term as 15 then third term will be 35
Common difference is 35 – 15 = 20
First term 15 – 20 = – 5
-5, 15, 35, 55
d) Sequence 4
If we choose second term as 12 then third term will be 38
Common difference is 38 – 12 = 26
First term 12 – 26 = -14
-14, 12, 38, 64
Question 2.
First term of an arithmetic sequence is 5 and sum of the first 6 terms is 105. Write the first six terms Answer:First term of an arithmetic sequence is 5 and sum of the first 6 terms is 105. Write the first six terms Six terms from the beginning are considered.
Answer:
Sum of first term and sixth term = \(\frac{105}{3}\) = 35
Given , first term = 5. So sixth term is 35 – 5 = 30
Sixth term – first term = 5 × common difference.
5 × common difference = 30 – 5 = 25,
Common difference = 5
5, 10, 15, 20, 25, 30
Question 3.
Sum of 7th and 8th terms of an arithmetic sequence is 50. Calculate the sum of first 14 terms.
Answer:
14 terms are considered.
Terms in the positions (1, 14), (2, 13), (3, 12), (4, 11), (5,10), (6,9), (7,8) gives the same sum, which is 50
Sum of first 14 terms = 50 × 7 = 350
Question 4.
Write the first three terms of each of the arithmetic sequences given below
a) The First term is 30 and the sum of first three terms 300
b) The First term is 30 and the sum of first four terms 300
c) The First term is 30 and the sum of the first five terms 300
d) The First term is 30 and the sum of first six ternis 300
Answer:
Write the first three terms of each of the arithmetic sequences given below
a) Second term = \(\frac{300}{3}\) = 100
Common difference is 70
Terms are 30, 100, 170
b) First term + fourth term = \(\frac{300}{2}\) = 150
Fourth term = 150 – 30 = 120
3 times common difference = 120 – 30 = 90
Common difference = \(\frac{90}{3}\) = 30
Terms are 30, 60, 90, 120
c) Third term = \(\frac{300}{5}\) = 60.
First term = 30
Common difference =15
Terms are 30, 45, 60, 75, 90
d) First term + sixth term = 100
Sixth term = 100 – 30 = 70
Five times common difference is 40, common difference is 8
30, 38, 46, 54, 62, 70
Arithmetic Sequences Class 10 Notes Pdf
Class 10 Maths Chapter 1 Arithmetic Sequences Notes Kerala Syllabus
Introduction
Sequence is the arrangement of numbers or objects in an order. The numbers or objects in the sequence are called terms of the sequence. In this chapter we discuss arithmetic sequence in its numerical form, terms of the sequence, properties of arithmetic sequence and its summation of terms.
Arithmetic Sequence
The sequence starting from a number and adding a number again and again is called arithmetic sequence. The number adding repeatedly is called common difference.
Arithmetic sequence has some specialities. Difference between any two terms is a multiple of common difference. In an arithmetic sequence, if odd number of consecutive terms are considered, middle term will be twice the sum of terms equidistant from it on both sides. Also the middle term will be obtained by dividing sum of terms by number of terms.
In this unit we learn arithmetic sequence without any algebraic thinking or formulae. Our thoughts on the sequence is based on the definition and properties of sequence.
→ Things arranged in definite positions as the first, second, third and so on is called sequence
Example : sequence of regular polygons △, □, ⬡, ⬠…………..
→ The arrangement of numbers obeying a definite rule which confirms its position as first, second , third… is called number sequence.
→ 1,4, 9, 16… is a number sequence. It is the square of natural numbers in the ascending order.
→ The sequence starting with a number and proceeding by adding a number again and again is called arithmetic sequence. The starting number is called first term and the number adding repeatedly is called common difference.
→ In general all sequences are related to the sequence of natural numbers or counting numbers. The numbers of the sequence in first, second , third positions are called terms of the sequence as 1st term, 2nd term, 3rd term and so on.
→ The difference between two terms of an arithmetic sequence is a multiple of common difference.
In any arithmetic sequence, the change in terms is the product of the change in position and a fixed number. In any arithmetic sequence, change in terms is proportional to the change in position. That is, in an arithmetic sequence, term difference is proportional to position difference.
Note that, common difference is the proportionality constant.
Arithmetic Sequences In an arithmetic sequence, the sum of the terms just before and just after it is twice this term.
→ In the arithmetic sequence sum of a term and the terms just before and after it is three times this term.
In three consecutive terms of an arithmetic sequence, middle term is \(\frac{1}{3}\) of sum of these terms
In five consecutive terms of an arithmetic sequence, the middle term is \(\frac{1}{5}\) of the sum of terms. That is, five times middle term is the sum of five terms.
This can be generalised for any odd number of terms.
→ In an arithmetic sequence, the sum of the two terms at the same distance behind and ahead a term, is twice this term
→ In an arithmetic sequence, the sum of a term and consecutive terms at the same distance behind and ahead, is the product of this term and the number of terms.
→ The sum of an odd number of consecutive terms of an arithmetic sequence is the product of the middle term and the number of terms.
→ In an arithmetic sequence, if the sum of two positions is equal to the sum of other two positions, then the sum of the terms at each pair is the same.
→ A sequence starting with a number and proceeding by adding one number again and again, is called an arithmetic sequence.
→ This number, got by subtracting the previous term from any term, is called the common difference of the arithmetic sequence.
→ In any arithmetic sequence, the change in terms is the product of the change in position and a fixed number.
→ In any arithmetic sequence, the change in terms is proportional to the change in position
→ In an arithmetic sequence, the sum of the two terms, at the same distance behind and ahead a term, is
twice this term.
→ In an arithmetic sequence, the sum of a term and the consecutive terms at the same distance behind and ahead, is the product of this term and the number of terms.
→ The sum of an odd number of consecutive terms of an arithmetic sequence is the product of the middle term and the number of terms.
→ In an arithmetic sequence, if one position is increased and another position is decreased by the same amount, the sum of the terms at these positions do not change.
→ In an arithmetic sequence, if the sum of two positions is equal to the sum of other two positions, then the sum of the terms at each pair is the same.