C1 Sequences

A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference d cm.

The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm.

Find

(b) the length of the shortest side of the polygon,
(c) the value of d.

Reply 1

Revelation

1

sum_16 = 72
n_16 = 6

I cant remember the equations, but you form 2 simultaneous equations from them^^ and you will find the unknowns d (difference) and a (first term).

Then you can easily do b and c putting numbers in the formula

Reply 2

ImperceptibleNinja

13

A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference d cm.

The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm.

Find

(b) the length of the shortest side of the polygon,

U[n] = a + (n-1)d
U[16] = 6 = a + 15d

S[n] = n/2 . (2a + (n-1)d)
S[16] = 16/2 . (2a + 15d) = 72
2a +15d = 9

[1]: a + 15d = 6
[2]: 2a +15d = 9 subtract

a = 3
so the shortest side is 3cm

(c) the value of d.

a + 15d = 6
15d = 6 - a = 3
d = 3/15 = 1/5
d = 0.2cm
[edit: just like Revelation said...]

Reply 3

evariste_3086

1

Fenchurch

A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference d cm.

The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm.

Find

(b) the length of the shortest side of the polygon,
(c) the value of d.

hint:
if the sides form a ap with largest term 6 and starting with shortest side then the sides also form an ap with first term 6 going down to the smallest side.
also use the fact the sum of the 16 sides=72 still holds considering the ap from largest to smallest and the answer follows.

edit: my hint is no good as the answer has been posted. It worked like this though:
original ap =a,a+2d,.....6
going other way we have
6.6+(-d),...,a
then
sum 16=(16)/2[12+15(-d)]=72
-15d=-3
ie d= 1/5

the 16th term of this ap is the first term of the original
ie
a=6+(15)(-1/5).........for the reversed ap d=-1/5
=3

Reply 4

danniella

4

Hello!

The longest side is a + 15d (the sixteenth term)
so a + 15d = 6

the perimeter is the sum to 16 terms of the arithmetic sequence,
so Sn = n/2 (2a + (n-1)d) = 72
72 = 8(2a + 15d)
9 = 2a+15d

So you have the simultaneous equations
a + 15d = 6 ------(1)
2a +15d = 9 ------(2)

subtract (1) from (2)
giving a=3

substitute that into equation (1)
3 + 15d = 6
15d = 3
d=1/5

(check in equation (2) :
8 (2x3 + 15x1/5) = 8(6 + 15/5)
= 8(6+3) = 8x9 = 72. Yay.)
so the shortest side is 3cm and d = 0.2

love danniella

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