# SCERT Kerala Maths Class 10/Arithmetic Sequences

SCERT Kerala Maths Class 10/ Arithmetic Sequences is about the textbook solutions of Class 10 Mathematics. Here you can find out practice problems for Class 10 Mathematics Arithmetic Sequences.

SCERT Kerala Maths Class 10 / Arithmetic Sequences – Chapter 1
Kerala State Board SSLC Maths Textbook Solutions
Arithmetic Sequences:
Important points to remember:

• A sequence got by starting with any number and adding a fixed number repeatedly is called an arithmetic sequence.
• An arithmetic sequence is a sequence in which we get the same number on subtracting from any term, the term immediately preceding it.
• The constant difference got by subtracting from any term the just previous term, is called the common difference of an arithmetic sequence.

1. Check whether each of the sequences given below is an arithmetic sequence. Give reasons. For the arithmetic sequences, write the common difference also.
i) Sequence of odd numbers
ii) Sequence of even numbers
iii) Sequence of fractions got as half the odd numbers
iv) Sequence of powers of 2
v) Sequence of reciprocals of natural numbers
Solution:
i). Sequence of odd numbers is 1, 3, 5, 7, 9—————-
The difference between consecutive terms are 3 – 1 = 2, 5 – 3 = 2
So the common difference = 2
Here the difference between the terms is a constant, so the given sequence is an arithmetic sequence.
ii) Sequence of even numbers is 2, 4, 6, 8, 10 ———————
The difference between consecutive terms are 4 – 2 = 2, 6 – 4 = 2
So the common difference = 2
Here the difference between the terms is a constant, so the given sequence is an arithmetic sequence.
iii) Sequence of fractions got as half the odd numbers is ½, 3/2, 5/2, 7/2 ————–
The difference between consecutive terms are 3/2 – ½ = 1, 5/2 – 3/2 = 1
So the common difference = 1
Here the difference between consecutive terms is a constant, so the given sequence is an arithmetic sequence.
iv) Sequence of powers of 2 is 2, 4, 8, 16 —————
The difference between consecutive terms are 4 – 2 = 2, 8 – 4 = 4
It does not have a common difference, so the given sequence is not an arithmetic sequence.
v) Sequence of reciprocals of natural numbers is 1, ½, 1/3, ¼, 1/5 —————–
The difference between consecutive terms are ½ – 1 = -1/2, 1/3 – ½ = -1/6
It does not have a common difference, so the given sequence is not an arithmetic sequence.
2. Look at the picture from the textbook (page no. 18)
If the pattern is continued, do the numbers of coloured squares form an arithmetic sequence? Give reasons.
Solution:
Sequence of the numbers of coloured squares in the given picture is 8, 12, 16 ————-
The difference between consecutive terms are 12 – 8 = 4, 16 – 12 = 4
So the common difference = 4
Here the difference between consecutive terms is a constant, so the given sequence is an arithmetic sequence.
3. See the pictures given in the textbook (page no 18)
i) How many small squares are there in each rectangle?
ii) How many large squares?
iii) How many squares in all?
Continuing this pattern, is each such sequence of numbers, an arithmetic sequence?
Solution:
i) Sequence of small squares in each rectangle is 2, 4, 6, 8 ————–
The difference between consecutive terms are 4 -2 = 2, 6 -4 = 2
So the common difference = 2
Here the difference between consecutive terms is a constant, so the given sequence is an arithmetic sequence.
ii) Sequence of large squares is 0, 1, 2, 3 ————–
The difference between consecutive terms are 1 – 0 = 0, 2 – 1 = 1
So the common difference = 1
Here the difference between consecutive terms is a constant, so the given sequence is an arithmetic sequence.
iii) Sequence of all the squares is 2, 5, 8, 11 ————–
The difference between consecutive terms are 5 – 2 = 3, 8 – 5 = 3
So the common difference = 3
Here the difference between consecutive terms is a constant, so the given sequence is an arithmetic sequence.
4. In the staircase shown here (for picture look at the textbook) the height of the first step is 10 centimetres and the height of each step after it is 17.5 centimetres.
i) How high from the ground would be someone climbing up, after each step?
ii) Write these numbers as a sequence.
Solution:
i) The height of the first step = 10 cm
The height of each step after it = 17.5 cm
The height of second step = 10 + 17.5 = 27.5 cm
The height of third step = 27.5 + 17.5 = 45 cm
The height of fourth step = 45 + 17.5 = 62.5 cm
The height of fifth step = 62.5 + 17.5 = 80 cm
The height of sixth step = 80 + 17.5 = 97.5 cm
ii) Sequence of height is 10, 27.5, 45, 62.5, 80, 97.5 ——————
5. In this picture (look at the textbook) the perpendiculars to the bottom line are equally spaced. Prove that, continuing like this, the lengths of perpendiculars form an arithmetic sequence.
Solution:
All the triangles given in the picture are similar right angled triangles.
Therefore the corresponding sides are in equal proportion. So if we take the length of the perpendicular of a smaller triangle as x, sequence of length of perpendicular will be x, 2x, 3x, ————
Here the common difference is x, so the perpendicular lengths are in an arithmetic sequence.
6. The algebraic expression of a sequence is xn = n3 – 6n2 + 13n – 7
Is it an arithmetic sequence?
Solution:
When n = 1, 1 – 6 + 13 – 7 = 1
When n = 2, 8 – 24 + 26 – 7 = 3
When n = 3, 27 – 54 + 39 – 7 = 5
When n = 4, 64 – 96 + 52 – 7 = 13
So the sequence is 1, 3, 5, 13, ———–
The difference between consecutive terms are 3 – 1 = 2, 5 – 3 = 2, 13 – 5 = 8
It does not have a common difference. So the given sequence is not an arithmetic sequence.