1. Topic-
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2. Content-
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| Introduction to Arithmetic Progressions |
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3. Objectives-
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1) To make students understand the mathematical facts.
2) To enhance problem solving skills in students.
3) To aware the students about Arithmetic Progressions.
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4. Materials and Aids-
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| Smart Class, White Board, Marker, Books etc. |
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5. Procedures/Methods-
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A. Introduction-
You must have observed that in nature, many things follow a certain
pattern, such as
the petals of a sunflower, the holes of a honeycomb, the grains on
a maize cob etc. We now look for some patterns which occur in our
day-to-day life. Some such
examples are :
(i) Reena applied for a job and got selected. She has been offered
a job with a starting monthly salary of Rs 8000, with an annual increment
of Rs 500 in her salary. Her salary (in Rs) for the 1st, 2nd, 3rd,
. . . years will be, respectively 8000, 8500, 9000, . . . .
(ii) The lengths of the rungs of a ladder decrease uniformly by 2
cm from bottom to top . The bottom rung is 45 cm in
length. The lengths (in cm) of the 1st, 2nd, 3rd, . . ., 8th rung
from the bottom to the top are, respectively
45, 43, 41, 39, 37, 35, 33, 31
In the examples above, we observe that the succeeding terms are obtained
by adding a fixed number.
In this chapter, we shall discuss one of these patterns in which succeeding
terms are obtained by adding a fixed number to the preceding terms.
We shall also see how
to find their nth terms and the sum of n consecutive terms, and use
this knowledge in
solving some daily life problems.
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B. Development-
an arithmetic progression is a list of numbers in which each term
is obtained by adding a fixed number to the preceding term except
the first term.
This fixed number is called the common difference (d) of the AP. Remember
that it can be positive, negative or zero.
In general form A.P. can be written as
a, a+d, a+2d, a+3d, "�"�"�"�.
We shall solve the questions given in exercise 5.1 in the book keeping
in mind all the concepts of A.P.
the nth term an of the AP with first term a and common difference
d is given by an = a + (n "� 1)
an is also called the general term of the AP. If there are m terms
in the AP, then am represents the last term which is sometimes also
denoted by l.
1) Determine the AP whose 3rd term is 5 and the 7th term is 9.
Solution : We have
a3 = a + (3 "� 1) d = a + 2d = 5 (1)
and a7 = a + (7 "� 1) d = a + 6d = 9 (2)
Solving the pair of linear equations (1) and (2), we get
a = 3, d = 1
Hence, the required AP is 3, 4, 5, 6, 7, . . .
2) How many two-digit numbers are divisible by 3?
Solution : The list of two-digit numbers divisible by 3 is :
12, 15, 18, . . . , 99
Is this an AP? Yes it is. Here, a = 12, d = 3, an = 99.
As an = a + (n "� 1) d,
we have 99 = 12 + (n "� 1) × 3
i.e., 87 = (n "� 1) × 3
i.e., n "� 1 =87/3
= 29
i.e., n = 29 + 1 = 30
So, there are 30 two-digit numbers divisible by 3.
We shall solve the questions given in exercise 5.2 up to question
no 6 in the book keeping in mind all the concepts of related to nth
term of A.P.
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