whats the answer to this?
write an equation fr an arithmetic sequence with a 1st term of 8 and 4th term of -7. im stuck.
In an arithmetic series, every term is separated from its neighbours
by the same difference.
So given an arithmetic sequence that starts with the first value A and
has a common difference of D, the first five terms are:
1st term: A
2nd term: A + D
3rd term: A + 2D
4th term: A + 3D
5th term: A + 4D
So the first five sums of the terms are:
1st sum: A
2nd sum: 2A + D
3rd sum: 3A + 3D
4th sum: 4A + 6D
5th sum: 5A + 10D
You know that the sum of the first 5 numbers is 500, so:
500 = 5A + 10D
and that the sum of the first 4 numbers is -8, so
-8 = 4A + 6D
Now you have two equations with two unknowns.
You can solve this and find the actual values of A and D. From those,
you can quickly write the first three numbers in the series and add
them up, or simply use the formula for the third sum: 3A + 3D
a = 8
a + 3d = -7
subtract them:
3d = -15
d = -5
So term(n) = a+(n-1)d
= 8 - 5(n-1)
= 13 - 5n
check for n = 1 and n = 4, it works
To find the equation for an arithmetic sequence, you need to identify the common difference (d).
The common difference (d) can be obtained by subtracting any term in the sequence from the next term.
In this case, you're given the 1st term (a₁) as 8 and the 4th term (a₄) as -7.
To find the common difference (d), subtract the 1st term from the 4th term:
d = a₄ - a₁ = (-7) - 8 = -15
Now that you have the common difference (d), you can write the equation for an arithmetic sequence:
aₙ = a₁ + (n - 1)d
Here, aₙ represents the nth term in the arithmetic sequence, a₁ is the first term, n is the position of the desired term, and d is the common difference.
Substituting the given values in the equation:
aₙ = 8 + (n - 1)(-15)
So, the equation for this arithmetic sequence is aₙ = 8 - 15(n - 1).