Question

A polygon has 25 sides the lenghts of which starting from the smallest sides are in AP.If the perimeter of the polygon is 2100cm and the lenght of the largest side is 20 times that of the smallest,then find the lenght of the smallest side and the common difference of the AP

Answer 1

Use sum of A.P. formula,

take n = 25,

smallest side = a,

largest side = a + (n - 1)d = 20a

sum = (n/2)*[2a+(n-1)d]

s = (n/2)*[a + a + (n - 1)d]

s = (n/2)*[a + 20a]

s = (n/2)*(21a)

Substitute values s = 2100, n = 25

(25/2)*21a = 2100

a = (2 X 2100)/(25 x 21) = 8 cm

smallest side is 8 cm

largest side = 20 x 8 = 160 cm

160 = 8 + (25 - 1)d

152 = 24d

d = 19/3 or 6.33

Answer 2

Answer:

The length of the smallest side is 8cm and the common difference of the arithmetic progression is 6.33.

Step-by-step explanation:

Given a number of sides in a polygon,  $n= 25$

Perimeter of the polygon is the sum of all sides, = 2100cm

The lengths starting from the smallest to the largest side are in arithmetic progression

The length of the largest side is 20 times that of the smallest.

Formula:

The sum of n terms in arithmetic progression is given by $S_{n} =\frac{n}{2} \big[2a+(n-1)d\big]$

where n is the number of terms, a is the first term and d is the difference between the terms.

Solution:

In the given problem, let the length of smallest side be $a$ and the largest length is $a_{n}$.

Substituting the values in the formula,

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

$\implies2100=\frac{25}{2} \big[2a+(25-1)d\big]$

$\implies2100=\frac{25} {2}\big[2a+24d\big]$

$\implies2100={25}(a+12d)$

$\implies a+12d=84$                                         ....(1)

The nth term in arithmetic progression is given by $a+(n-1)d$

Therefore, 25th side is the largest length, $a_{n} =a+(25-1)d=a+24d$

Given the length of the largest side is 20 times that of the smallest, therefore $a_{n} =20a$.

$\therefore a+24d=20a$

$\implies 19a=24d$

$\implies d=\frac{19a}{24}$

Substituting value of $d$ in equation (1),

$a+12\times \frac{19a}{24}=84$

$\implies 21a=168$

$\implies a=\frac{168}{21}=8$

And hence $d=\frac{19a}{24}=\frac{19\times 8}{24}\frac{19}{3}=6.33$

Hence, the length of the smallest side is 8cm and the common difference of the AP is 6.33.