Name: Applications of Pascal's Triangle
Uploaded: 2019-07-19
Duration: 3 min 57 s
Description: What does the inside of a tesseract look like? Pascal's Triangle can tell us. Watch this video to know more about Applications of Pascal's Triangle

Question

In the following figures, polygons are formed by increasing the number of sides using matchsticks as given below.

Find the number of sticks required to form the next three shapes by tabulation and generalisation.

Accepted Answer

In the above pattern of polygons, in the first shape $(x = 1)$ , we get a closed shape called a triangle. Similarly the second shape $(x = 2)$ gives a four sided polygon and the third shape $(x = 3)$ is a five sided polygon and continuing in the same way two more shapes are formed. If the number of match sticks required to form each of the shapes is taken as y, then the values of $x and y$ are tabulated as given below.

$X$ $1$ $2$ $3$ $4$ $5$ ...

$Y$ $3$ $4$ $5$ $6$ $7$ ...

Observe the table and express y in terms of $x$ as below :

when $x = 1, y = 3 = 1 + 2$
when $x = 2, y = 4 = 2 + 2$
when $x = 3, y = 5 = 3 + 2$
when $x = 4, y = 6 = 4 + 2$
when $x = 5, y = 7 = 5 + 2$

Hence, each of the values of $y$ which we get from the table is $2$ more than $x$ . That is $y = x + 2$ .

Therefore,

6th shape $(x = 6)$ will have $y = 8 = 6 + 2$ ( 8 match sticks)
7th shape $(x = 7)$ will have $y = 9 = 7 + 2$ ( 9 match sticks)
8th shape $(x = 8)$ will have $y = 10 = 8 + 2$ (10 match sticks)

We clearly see that the next three shapes will require $8, 9 and 10$ matchsticks.

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