A piece of string of length 5m long is cut into n pieces in such a way that the lengths of the pieces are an arithmetic sequence. If the lengths of the longest and the shortest pieces are 1m and 25cm respectively, calculate n.
In an arithmetic progression:
an = a + ( n - 1 ) d
where
a = the initial term
d = the common difference of successive members
an = the nth term
Lengths shortest pieces:
a1 = a + ( 1 - 1 ) d = a + 0 ∙ d = a
a1 = a = 25 cm
Lengths longest pieces:
an = 1 m = 100 cm
an = a + ( n - 1 ) d
100 = 25 + ( n - 1 ) d
The sum of n terms of an arithmetic progression:
Sn = ( n / 2 ) [ 2 a + ( n -1 ) d ]
In this case a = 25 cm so:
Sn = ( n / 2 ) [ 2 ∙ 25 + ( n -1 ) d ]
Sn = ( n / 2 ) [ 50 + ( n -1 ) d ]
The sum of n terms of this arithmetic progression is 5 m
Sn = 5 m = 500 cm
500 = ( n / 2 ) [ 50 + ( n -1 ) d ]
Now you must solve system of two equations:
25 + ( n - 1 ) d = 100
( n / 2 ) [ 50 + ( n -1 ) d ] = 500
The solution is:
d = 75 / 7 , n = 8
Your arithmetic progression:
a1 = 25
a2 = 25 + 75 / 7 = 175 / 7 + 75 / 7 = 250 / 7
a3 = 250 / 7 + 75 / 7 = 325 / 7
a4 = 325 / 7 + 75 / 7 = 400 / 7
a5 = 400 / 7 + 75 / 7 = 475 / 7
a6 = 475 / 7 + 75 / 7 = 550 / 7
a7 = 550 / 7 + 75 / 7 = 625 / 7
a8 = 625 / 7 + 75 / 7 = 700 / 7 = 100
You can check the sum of this arithmetic progression.
a1 + a2 + a3 + a4 + a5 + a6 + a7 =
25 + 250 / 7 + 325 / 7 + 400 / 7 + 475 / 7 + 550 / 7 + 625 / 7 + 100 =
25 + ( 250 / 7 + 325 / 7 + 400 / 7 + 475 / 7 + 550 / 7 + 625 / 7 ) + 100 =
25 + 2625 / 7 + 100 = 25 + 375 + 100 = 500
The sum of this arithmetic progression = 500 cm
My little typo.
the sum of this arithmetic progression is:
a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 =
25 + 250 / 7 + 325 / 7 + 400 / 7 + 475 / 7 + 550 / 7 + 625 / 7 + 100
...
Ah, the ol' arithmetic sequence with a string twist! Let's unravel this mathy enigma, shall we?
First, let's convert the lengths of the longest and shortest pieces to the same units. We have 1m for the longest piece and 25cm for the shortest piece. Since 1 meter is equal to 100 centimeters, that means the shortest piece is 0.25 meters long.
Now we know that the longest piece is 1 meter and the shortest piece is 0.25 meters. We can assume that the common difference, let's call it 'd', is the same for all the pieces in the sequence.
Using the formula for the nth term of an arithmetic sequence, which is an = a1 + (n - 1) * d, we can plug in the values we have: a1 = 0.25, an = 1.
1 = 0.25 + (n - 1) * d
Simplifying a bit:
0.75 = (n - 1) * d
Now, since the lengths form an arithmetic sequence, we know that the total length of the string is the sum of the lengths of all the pieces. The sum of an arithmetic series is given by the formula Sn = n/2 * (a1 + an), where Sn is the sum of the series and n is the number of terms.
Substituting the values we have, we get:
5 = n/2 * (0.25 + 1)
Simplifying that equation:
10 = n * (0.25 + 1)
10 = n * 1.25
Dividing both sides of the equation by 1.25 gives us:
n = 10 / 1.25
n = 8
So, according to my calculations, there are 8 pieces in total. Voila!
To solve this problem, we need to find the common difference of the arithmetic sequence and then use that to determine the number of pieces.
Let's start by converting the lengths of the longest and shortest pieces to the same unit.
The longest piece is 1m and the shortest piece is 25cm.
Since 1m = 100cm, the longest piece is 100cm and the shortest piece is 25cm.
Let's assume the common difference is d.
So, the lengths of the pieces will be:
25cm, 25cm + d, 25cm + 2d, 25cm + 3d, ...
The sum of an arithmetic sequence is given by the formula:
Sum = (n/2)(first term + last term)
In our case, the first term is 25cm and the last term is 100cm.
So, the sum of the arithmetic sequence is:
5m = (n/2)(25cm + 100cm)
Let's convert 5m to centimeters:
5m = 500cm
Now, we can solve for n:
500cm = (n/2)(25cm + 100cm)
500cm = (n/2)(125cm)
500cm = 125n
n = 500cm / 125cm
n = 4
Therefore, the number of pieces, n, is 4.
To solve this problem, we need to find the number of pieces, which is represented by n.
We are given that the lengths of the longest and shortest pieces are 1m and 25cm respectively. Let's convert the lengths to a common unit, meters.
1m = 1m
25cm = 0.25m (since 1m = 100cm, then 1cm = 1/100m, and 25cm = 25/100m = 0.25m)
Now, let's set up the arithmetic sequence where the lengths of the pieces form a sequence. We know that the longest piece is 1m and the shortest piece is 0.25m. Let d be the common difference between the lengths.
So, the lengths of the pieces can be represented as: 0.25m, 0.25m + d, 0.25m + 2d, ..., 0.25m + (n - 1)d
We are also given that the total length of the string is 5m, so we can set up the equation:
0.25m + 0.25m + d + 0.25m + 2d + ... + 0.25m + (n - 1)d = 5m
Now let's simplify the equation:
(0.25m + 0.25m + ... + 0.25m) + (d + 2d + ... + (n - 1)d) = 5m
The first part of the equation can be simplified as (0.25m) * n.
The second part of the equation can be simplified using the formula for the sum of an arithmetic series:
Sum = (n/2) * (first term + last term)
Sum = (n/2) * (d + 0.25m + (n - 1)d)
Substituting these back into the equation, we have:
(0.25m) * n + (n/2) * (d + 0.25m + (n - 1)d) = 5m
Now, we have an equation with two variables, n and d. We need to solve for n.
To do this, we need another equation that relates n and d. We know that the longest and shortest pieces have lengths of 1m and 0.25m respectively. We can set up this equation:
1m = 0.25m + (n - 1)d
Now, we have a system of two equations with two variables. We can solve this system to find the values of n and d.
By substituting the second equation into the first equation, we have:
(0.25m) * n + (n/2) * (0.25m + (1m - 0.25m)) = 5m
Simplifying further:
(0.25m) * n + (n/2) * (0.25m + 0.75m) = 5m
(0.25m) * n + (n/2) * (1m) = 5m
Now, we have an equation with only one variable, n. We can solve for n by isolating it on one side of the equation:
(0.25m) * n + (n/2) * (1m) = 5m
Multiply every term by 2 to eliminate the fraction:
(2 * 0.25m) * n + n * (1m) = 2 * 5m
0.5m * n + n * m = 10m
Multiply every term by m to eliminate the fraction:
0.5m * n + n * (m) = 10m * (m)
Simplify further:
0.5mn + mn = 10m^2
Combine like terms:
1.5mn = 10m^2
Divide both sides by 1.5m:
n = (10m^2) / (1.5m)
Simplify:
n = 20m / 3
Therefore, the value of n is 20m/3.