The 5th term and the 8th term of an arithmetic sequence are 18 and 27 respctively.
a)Find the 1st term and the common difference of the arithmetic sequence.
b)Find the general term of the arithmetic sequence.
If a sequence has a first term of 16 and a common difference of -7, what is the 5th and 6th term
a) To find the first term and the common difference of the arithmetic sequence, we need to use the formula for the nth term of an arithmetic sequence:
An = A1 + (n-1)d
Where An is the nth term, A1 is the first term, and d is the common difference.
We are given the following information:
A5 = 18
A8 = 27
Let's use these values to set up two equations:
A5 = A1 + 4d = 18 (1)
A8 = A1 + 7d = 27 (2)
Now we can solve these equations simultaneously.
Subtract (1) from (2) to eliminate A1:
A8 - A5 = (A1 + 7d) - (A1 + 4d)
27 - 18 = 7d - 4d
9 = 3d
d = 3
Substitute the value of d back into equation (1) to find A1:
18 = A1 + 4(3)
18 = A1 + 12
A1 = 18 - 12
A1 = 6
Therefore, the first term (A1) is 6 and the common difference (d) is 3.
b) Now that we know the first term and the common difference, we can find the general term of the arithmetic sequence.
The formula for the nth term is:
An = A1 + (n-1)d
Plugging in the values we found:
An = 6 + (n-1)(3)
An = 6 + 3n - 3
An = 3n + 3
So, the general term of the arithmetic sequence is An = 3n + 3.
To find the 1st term (a₁) and the common difference (d) of an arithmetic sequence, we can use the formula:
aₙ = a₁ + (n - 1)d
where aₙ represents the nth term of the sequence.
a) Find the 1st term and the common difference of the arithmetic sequence:
Given that the 5th term (a₅) is 18 and the 8th term (a₈) is 27, we can set up two equations using the formula mentioned above:
1st equation: a₅ = a₁ + (5 - 1)d
Substituting a₅ = 18 and solving for a₁ + 4d = 18
2nd equation: a₈ = a₁ + (8 - 1)d
Substituting a₈ = 27 and solving for a₁ + 7d = 27
Now we have a system of two equations with two unknowns (a₁ and d). We can solve this system of equations using substitution or elimination method.
From the first equation, we have a₁ = 18 - 4d.
Substituting this value of a₁ into the second equation, we get:
(18 - 4d) + 7d = 27
18 + 3d = 27
3d = 27 - 18
3d = 9
d = 9/3
d = 3
Now substitute the value of d back into the first equation to find a₁:
a₁ = 18 - 4(3)
a₁ = 18 - 12
a₁ = 6
So the 1st term (a₁) of the arithmetic sequence is 6 and the common difference (d) is 3.
b) Find the general term (aₙ) of the arithmetic sequence:
To find the general term (aₙ), we can use the formula mentioned earlier:
aₙ = a₁ + (n - 1)d
Substituting the values of a₁ and d we found, the general term becomes:
aₙ = 6 + (n - 1)3
aₙ = 6 + 3n - 3
aₙ = 3n + 3
Therefore, the general term of the arithmetic sequence is aₙ = 3n + 3.
13,26,39,52
common difference=3,1st term=6and general term=3+3n
an = a1 + (n-1)d
18 = a1 + (5-1)d
27 = a1 + (8-1)d
so
-9 = -3d
d = 3
then a1 = 18 - 12 = 6
so
an = 6 + (n-1)3