Maharashtra Board Class 10 Maths Part I Chapter 3 Arithmetic Progression PDF Download

Arithmetic Progression

Sequence

We write numbers 1, 2, 3, 4, . . . in an order. In this order we can tell the position of any number. For example, number 13 is at 13th position. The numbers 1, 4, 9, 16, 25, 36, 49, . . . are also written in a particular order. Here 16 = 4² is at 4th position. Similarly, 25 = 5² is at the 5th position; 49 = 7² is at the 7th position. In this set of numbers also, place of each number is determined.

A set of numbers where the numbers are arranged in a definite order, like the natural numbers, is called a sequence.

In a sequence a particular number is written at a particular position. If the numbers are written as a₁, a₂, a₃, a₄ . . . then a₁ is first, a₂ is second, . . . and so on. It is clear that a_n is at the nth place. A sequence of the numbers is also represented by alphabets f₁, f₂, f₃, . . . and we find that there is a definite order in which numbers are arranged.

When students stand in a row for drill on the playground they form a sequence.

We have experienced that some sequences have a particular pattern.

Complete the given pattern:

Complete the given pattern:

Look at the patterns of the numbers. Try to find a rule to obtain the next number from its preceding number. This helps us to write all the next numbers.

See the numbers 2, 11, -6, 0, 5, -37, 8, 2, 61 written in this order.

Here a₁ = 2, a₂ = 11, a₃ = -6, . . . This list of numbers is also a sequence. But in this case we cannot tell why a particular term is at a particular position; similarly we cannot tell a definite relation between the consecutive terms.

In general, only those sequences are studied where there is a rule which determines the next term.

For example (1) 4, 8, 12, 16 . . . (2) 2, 4, 8, 16, 32, . . . (3) \(\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}\) . . .

Terms In A Sequence

In a sequence, ordered terms are represented as t₁, t₂, t₃, . . . .t_n . . . In general sequence is written as {t_n}. If the sequence is infinite, for every positive integer n, there is a term t_n.

Activity I : Some sequences are given below. Show the positions of the terms by t₁, t₂, t₃, . . .

(1) 9, 15, 21, 27, . . . Here t₁ = 9, t₂ = 15, t₃ = 21, . . .

(2) 7, 7, 7, 7, . . . Here t₁ = 7, t₂ = , t₃ = , . . .

(3) -2, -6, -10, -14, . . . Here t₁ = -2, t₂ = , t₃ = , . . .

Activity II : Some sequences are given below. Check whether there is any rule among the terms. Find the similarity between two sequences.

To check the rule for the terms of the sequence look at the arrangements on the next page, and fill the empty boxes suitably.

(1) 1, 4, 7, 10, 13, . . . (2) 6, 12, 18, 24, . . .

(3) 3, 3, 3, 3, . . . (4) 4, 16, 64, . . .

(5) -1, -1.5, -2, -2.5, . . . (6) 13, 23, 33, 43, . . .

Teacher's Note

Numbers arranged in order are called sequences. Like your roll number in class, you have a fixed position in the line during morning assembly.

Exam Trick

Remember: A sequence has numbers in a fixed order. Just like your school roll numbers are in order, sequences are also in order with a pattern.

Points to Remember

A sequence is a set of numbers in a definite order.
Each number in a sequence has a fixed position.
Some sequences follow a special pattern or rule.
We can find any number in a sequence if we know the pattern.

This is a preview of the first 3 pages. To get the complete book, click below.