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Algebraic Expressions

Understand Factors and Coefficients of terms; Like and Unlike Terms, etc

Let us take some matchsticks and join them to form some patterns as shown in the following figure. The number of matchsticks used to make each pattern is written alongside it.

Now, let us try to represent the number of matchsticks used to form each pattern with the help of an algebraic pattern. The given pattern is formed by repeating the shape, which is made of 5 matchsticks.

Thus, we require 5 matchsticks to form the first shape, 8 matchsticks to form the second shape, 11 matchsticks to form the third shape, and so on. Thus, for each extra shape, we require 3 more matchsticks.

Now, we can represent the numbers 5, 8, 11, and 14 as expressions involving the numbers 1, 2, 3, and 4 as

5 = 3 × 1 + 2

8 = 3 × 2 + 2

11 = 3 × 3 + 2

14 = 3 × 4 + 2

Thus, we require (3n + 2) number of matchsticks to form n number of shapes.

We can verify this expression by taking n = 1, 2, 3 ….

If we take n = 3, then (3n + 2) = 3 × 3 + 2 = 9 + 2 = 11, which is indeed the number of line segments required to make the required shape.

Using this expression (3n + 2), we can calculate the number of line segments required to make any number of shapes. For example, if we want to make 25 such shapes, then we require (3n + 2) = 3 × 25 + 2 = 75 + 2 = 77 matchsticks.

If we actually draw the pattern and count the number of matchsticks, then it will be very time consuming. But the use of algebraic expression has made our job much simpler.

We come across these types of number patterns in Algebra.

Let us discuss one such number pattern, which does not involve any figure.

4, 13, 22, 31, 40 … 9n − 5

The term which occurs at the nth position in the pattern is given by the expression 9n − 5. If we want to find the number at the 50th position in the pattern, then we can easily calculate it by substituting n = 50 in this expression.

Thus, number at the 50th position in the pattern = 9 × 50 − 5 = 450 − 5 = 445

Similarly, the number in 91st position is 9 × 91 − 5 = 819 − 5 = 814

Let us discuss some more examples based on number patterns.

Example 1:

Observe the patterns of triangles made from matchsticks. Find the algebraic

expression that gives the number of matchsticks in terms of the number of triangles.

Find the number of matchsticks required to make 29 such triangles. Also find the number of triangles formed, if there are 51 matchsticks.

Solution:

The given pattern is formed by repeating the shape that is made from 3 matchsticks. The number of matchsticks required to form 1, 2, 3, 4 ... shapes are 3, 5, 7, 9 ... respectively. Her…

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