Question

Write the general formula for the following pattern​

Answer 1

In the 1st figure there are 6 matchsticks in total. It consists of a triangle and a square combined at one matchstick.

A triangle is made by 3 matchsticks and a square is made by 4 matchsticks.

So the 6 matchsticks can be obtained as,

$\longrightarrow3+4-1=6$

$\longrightarrow6+5\times0=6$

$\longrightarrow6+(1-1)5=6\quad\quad\dots(1)$

The 2nd figure is obtained by combining twice the 1st figure, combined at one matchstick.

Here are 11 matchsticks which can be obtained as,

$\longrightarrow 2\times6-1=11$

$\longrightarrow (6+6)-1=11$

$\longrightarrow 6+5\times1=11$

$\longrightarrow 6+(2-1)5=11\quad\quad\dots(2)$

The 3rd figure is obtained by combining 1st and 2nd figures, combined at one matchstick.

Here are 16 matchsticks which can be obtained as,

$\longrightarrow6+11-1=16$

$\longrightarrow6+10=16$

$\longrightarrow6+(3-1)5=16\quad\quad\dots(3)$

From (1), (2) and (3), we can observe that no. of matchsticks in each figure form an AP of first term 6 and common difference 5.

• $a=6$

• $d=5$

So the no. of matchsticks in n'th figure will be,

$\longrightarrow a_n=a+(n-1)d$

$\longrightarrow a_n=6+(n-1)5$

$\longrightarrow a_n=6+5n-5$

$\longrightarrow\underline{\underline{a_n=5n+1}}$