Write three arithmetic sequences with 30 as the sum of the first five terms
sum(5) = (5/2)(2a + 4d) = 30
2a + 4d = 12
a + 2d = 6
a = 6-2d
so we can assign any value to d
e.g. d = 7
then a = 6-14 = -8
one possible sequence is:
-8 , -1, 6, 13, 20
(their sum is 30)
let d = 3
a = 6-6 = 0
terms are:
0, 3, 6, 9, 12
(their sum is 30)
your turn.
Take first sequence as
x, x+1, x+2, x+3, x+4,
5x+10=30
5x=30-10
5x=20
x=20/5=4
Substitute the value of x
Then we get
4, 5, 6, 7, 8
Do the following like this
Then we get d= 1
Take second sequence as x, x+2, x+4, x+6, x+8
Then we get d = 2
Take third as x, x+3, x+6, x+9, x+12
Then we get d=3
We get the three are
4, 5, 6, 7, 8, 9
2, 4, 6, 8, 10
0, 3, 6, 9, 12
Write four arithmetoc sequences with 100 as the sum of the first four terms.
To create arithmetic sequences with a sum of 30 as the sum of the first five terms, we need to find three different sequences that contain five terms whose sum is equal to 30.
Let's start by finding the common difference (d) of the arithmetic sequence that we will use to create the sequences.
The formula to find the sum of the first "n" terms of an arithmetic sequence is: Sum = (n/2)(2a + (n-1)d), where "a" is the first term and "d" is the common difference.
In this case, we know the sum is 30 and the number of terms is 5. So we can rearrange the formula to solve for the common difference:
30 = (5/2)(2a + (5-1)d)
30 = (5/2)(2a + 4d)
30 = (5/2)(2a + 4d)
30 = 10a + 20d
We need to find an "a" and "d" that satisfy this equation. We can rearrange it to solve for "d":
30 - 10a = 20d
d = (30 - 10a) / 20
d = (3 - a) / 2
Now we can choose a value for "a" and calculate the corresponding value of "d" to create different arithmetic sequences with a sum of 30:
Arithmetic Sequence 1:
Choose a = 2
Calculate d:
d = (3 - 2) / 2 = 1/2
The sequence becomes: 2, 2.5, 3, 3.5, 4
Arithmetic Sequence 2:
Choose a = 5
Calculate d:
d = (3 - 5) / 2 = -1
The sequence becomes: 5, 4, 3, 2, 1
Arithmetic Sequence 3:
Choose a = 0
Calculate d:
d = (3 - 0) / 2 = 1.5
The sequence becomes: 0, 1.5, 3, 4.5, 6
So, the three arithmetic sequences with a sum of 30 as the sum of the first five terms are:
1) 2, 2.5, 3, 3.5, 4
2) 5, 4, 3, 2, 1
3) 0, 1.5, 3, 4.5, 6
Let d=4 then sequence is
-2,2,6,10,14