arithmetic series

**sigma1** · December 8th 2010, 03:11 PM

A piece of cord is of length 64 m. It is cut into 16pieces whose lengths are in arithmetic
progression. The length of the longest piece is three times that of the smallest. Find the length of the shortest piece of chord.

$S_n = 64$

length of shortest piece is $x$
length of longest piece is $3x$

is it right to assume that the shortest piece is the first term and the largest piece is the last term.

is this correct?
i have formed two equations

$3x = x+(15)d$ equating the longest piece to the last term

$64 = 16/8 (x + x+15d)$ equating the total length of cord.

**DrSteve** · December 8th 2010, 03:47 PM

Why are you dividing by 8? I believe you should be dividing by 2. Also, replace x+15d by 3x in the second equation.

**pickslides** · December 8th 2010, 03:51 PM

Here's some hints to get you started

$S_{n} = n\times \frac{a+t_n}{2}$

$S_{16} = 64$

$a+(a+d)+(a+2d)+\dots +(a+15d)=64$

**Soroban** · December 8th 2010, 03:56 PM

Hello, sigma1!

Your set-up is correct . . . except for that "8".

$\text{A piece of cord is of length 64 m.}$
$\text{It is cut into 16 pieces whose lengths are in arithmetic progression.}$
$\text{The length of the longest piece is three times that of the smallest.}$
$\text{Find the length of the shortest piece of cord.}$

$S_n = 64$

$\text{length of shortest piece is }x$
$\text{length of longest piece is }3x$

$\text{Is it right to assume that the shortest piece is the first term}$ . . $\text{and the largest piece is the last term?}$

$\text{i have formed two equations:}$

$3x \:=\: x+(15)d\qquad \text{equating the longest piece to the last term}$

$64 \:= \frac{16}{2}(2x +15d) \qquad\text{equating the total length of cord}$

$\begin{array}{cccccccc}<br /> \text{Your 1st equation is:} & 2x - 15d &=& 0 & [1] \\<br /> \text{Your 2nd equaton is:} & 2x + 15d &=& 8 & [2] \end{array}$

Subtract [2] - [1]: . $30d \:=\:8 \quad\Rightarrow\quad d \:=\:\frac{4}{15}$

Substitute into [1]: . $2x - 15(\frac{4}{15}) \:=\:0 \quad\Rightarrow\quad x \:=\:2$

Therefore, the shortest piece is $2\,m.$

**sigma1** · December 8th 2010, 04:07 PM

Originally Posted by Soroban

Hello, sigma1!

Your set-up is correct . . . except for that "8".

$\begin{array}{cccccccc}<br /> \text{Your 1st equation is:} & 2x - 15d &=& 0 & [1] \\<br /> \text{Your 2nd equaton is:} & 2x + 15d &=& 8 & [2] \end{array}$

Subtract [2] - [1]: . $30d \:=\:8 \quad\Rightarrow\quad d \:=\:\frac{4}{15}$

Substitute into [1]: . $2x - 15(\frac{4}{15}) \:=\:0 \quad\Rightarrow\quad x \:=\:2$

Therefore, the shortest piece is $2\,m.$

ok thanks for the corrections. the 8 was a mistake i was wrighting the result instead of the 2 that should have been dividing it..

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