Question

A polygon has 25 sides, the length of which starting from the smallest side are in arithmetical progression. If the perimeter of the polygon is 2100 cm and the length of the largest side is 20 times that of the smallest side, find the length of the smallest side and common difference of the arithmetic progression.

Answer 1

Answer:

Length of the smallest side of the polygon is 8 cm

common difference is $\frac{19}{3}$

Step-by-step explanation:

Formula used:

The n th term of the A.P a, a+d, a+2d, .......

is $t_n=a+(n-1)d$

The sum of n terms of an A.P a, a+d, a+2d,.... is

$S_n=\frac{n}{2}[2a+(n-1)d]$

Let the smallest side of the polygon be a

and the common difference be d

As per given data,

Length of the largest side = 20*(smallest side)

That is,

$t_{25}=20*t_1$

$a+24d=20a$

$24d=19a$....................(1)

Perimeter of the polygon = 2100 cm

That is

$S_{25}=2100$

$\frac{n}{2}[2a+(n-1)d]=2100$

$\frac{25}{2}[2a+24d]=2100$

$\frac{25}{2}[2a+19a]=2100$

$\frac{25}{2}[21a]=2100$

$\frac{25a}{2}=100$

$25a=100*2$

$25a=200$

$a=\frac{200}{25}$

$a=8$

put a=8 in (1), we get

$24d=19(8)$

$d=\frac{19(8)}{24}$

$d=\frac{19}{3}$