Question

Racheal is making patterns with match sticks. How many crosses would the pattern have when 65 match sticks are used?

Answer 1

Answer:

16

Step-by-step explanation:

On observing. the pattern:

• When 5 matchsticks are used, the number of crosses obtained is = 1
• When 9 matchsticks are used, the number of crosses obtained is = 2
• When 13 matchsticks are used, the number of crosses obtained is = 3

Look closely at the number of matchsticks that we're using!

5, 9, 13

If we subtract 5 from 9, I.e., 9 - 5

= 4

If we subtract 9 from 13, I.e., 13 - 9

= 4

Woah! This "4" is best described as a common difference.

So, the pattern we observed above is not any common pattern but an Arithmetic Progression, where the number of crosses is equivalent to the term number and the number of matchsticks being used is equivalent to the arithmetic term.

For the given Arithmetic Progression:

• a1 (first term) = 5
• common difference (d) = 4

We have to, now, find the number of crosses that will be formed when 65 matchsticks are in use, that is,

the term number (n) of the AP for which the arithmetic term is 65 (An)

$\boxed{ \mathsf{a _{n} = a + (n - 1)d }}$

$\implies \mathsf{65= 5+ (n - 1)4 }$

$\implies \mathsf{65 - 5= (n - 1)4 }$

$\implies \mathsf{60= (n - 1)4 }$

$\implies \mathsf{15= (n - 1) }$

$\implies \mathsf{n = 15 + 1}$

$\implies \mathsf{ \underline {n = 16}}$

Answer:

By using 65 matchsticks, we'll get 16 crosses.

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Hope this helps!