Question

We know the sum of the interior angles of a triangle is 180°. Show that the sums

of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic

progression. Find the sum of the interior angles for a 21 sided polygon.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

Sum of the interior angles of a triangle is 180°.

It means sum of interior angles with 3 sides is 180°

Now, Sum of interior angles of a quadrilateral is equals to sum of interior angles of two triangles = 2 × 180° = 360°.

[ See the attachment ]

Now, Sum of interior angles of a pentagon is equals to sum of interior angles of three triangles = 3 × 180° = 540°.

[ See the attachment ]

Now, Sum of interior angles of a hexagon is equals to sum of interior angles of two triangles = 4 × 180° = 720°.

[ See the attachment ]

So, We get the sum of interior angles series as

$\rm :\longmapsto\:180, \: 360, \: 540, \: 720, \: - - - - -$

Here,

$\red{\rm :\longmapsto\:a_1 = 180}$

$\red{\rm :\longmapsto\:a_2 = 360}$

$\red{\rm :\longmapsto\:a_3 = 540}$

$\red{\rm :\longmapsto\:a_4 = 720}$

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.

.

.

.

So, Consider

$\red{\rm :\longmapsto\:a_2 - a_1 = 360 - 180 = 180}$

$\red{\rm :\longmapsto\:a_3 - a_2 = 540 - 360 = 180}$

$\red{\rm :\longmapsto\:a_4 - a_3 = 720 - 540 = 180}$

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.

.

.

$\rm \implies\:a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = - -$

$\bf\implies \:180, \: 360, \: 540, \: 720, \: - - - -forms \: an \: AP$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

To find the sum of all interior angles of a polygon having 21 sides, we have to find 19th term of an AP series.

First term is sum of interior angles of 3 sides

Second term is sum of interior angles of 4 sides.

Thus

$\rm :\longmapsto\:a_{19} = a + (19 - 1)d$

$\rm :\longmapsto\:a_{19} = a + 18d$

$\rm :\longmapsto\:a_{19} = 180 + 18 \times 180$

$\rm :\longmapsto\:a_{19} = 180 + 3240$

$\rm \implies\:\boxed{ \tt{ \: a_{19} \: = \: 3420 \: }}$

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More to know :-

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.