Arithmetic sequences
Find a8 and an for the sequence: -3, -7,-11
the common difference is -4
an = 1 - 4n
What is a8 ? Could it possibly be -31. Do i plug in 8 for the equation an=1-4n
Scott has already given you the rule which is an=1-4n.
To check the rule,
put n=1, an=1-4=-3 (works)
put n=2, an=1-4(2)=-7 (works)
To check a8, you can either use the given rule, or you can numerate the terms (8 is not too bad!)
-3, -7, -11, -15.....
well, let's try to take this step by step. Let's call the two lower vertices L and R. Suppose we have
π/3
L = (h,-k)
Then the line LC has slope
mLC = k/(1-h)
y+k = k/(1-h) (x-h)
Since LRC is equilateral, its angles are all π/3. So, the slope of LR is
mLR = tan(arctan(k/(1-h))-π/3)
and the equation of LR is
y+k = mLR (x-h)
Similarly, line RC has slope
mRC = tan(arctan(mLR)-π/3)
and its equation is (using point B)
y = tan(arctan(mLR)-π/3) (x-2)
well, let's try to take this step by step. Let's call the two lower vertices L and R. Suppose we have
L = (h,-k)
Then the line LC has slope
mLC = k/(1-h)
and its equation (using point A) is
y = k/(1-h) (x-1)
Since LRC is equilateral, its angles are all π/3. So, the slope of LR is
mLR = tan(arctan(k/(1-h))-π/3)
and the equation of LR is
y+k = mLR (x-h)
Similarly, line RC has slope
mRC = tan(arctan(mLR)-π/3)
and its equation is (using point B)
y = tan(arctan(mLR)-π/3) (x-2)
well, let's try to take this step by step. Let's call the two lower vertices L and R. Suppose we have
L = (h,-k)
Then the line LC has slope
mLC = k/(1-h)
and its equation (using point A) is
y = k/(1-h) (x-1)
Since LRC is equilateral, its angles are all π/3. So, the slope of LR is
mLR = tan(arctan(mLC)-π/3)
and the equation of LR is
y+k = mLR (x-h)
Similarly, line RC has slope
mRC = tan(arctan(mLR)-π/3)
and its equation is (using point B)
y = tan(arctan(mLR)-π/3) (x-2)
To find a8 and an for the arithmetic sequence -3, -7, -11, we need to determine the common difference and the general formula for the nth term of the sequence.
In an arithmetic sequence, each term is formed by adding a constant value called the common difference (d) to the previous term.
To find the common difference, we subtract any term from its previous term. Let's choose -7 and -3:
-7 - (-3) = -7 + 3 = -4
So, the common difference (d) is -4.
The formula for the nth term in an arithmetic sequence is:
an = a1 + (n - 1)d
where:
an represents the nth term,
a1 is the first term of the sequence, and
d is the common difference.
Now, we can find a8 and an:
1. Finding a8:
Using the formula, we substitute n = 8:
a8 = a1 + (8 - 1)d
= a1 + 7d
= -3 + 7(-4)
= -3 - 28
= -31
So, a8 = -31.
2. Finding an:
Since we don't have the specific value for n, we can use the formula to represent the general nth term:
an = a1 + (n - 1)d
So, an = -3 + (n - 1)(-4)
= -3 - 4n + 4
= 1 - 4n
So, an = 1 - 4n.
Therefore, a8 = -31 and an = 1 - 4n for the arithmetic sequence -3, -7, -11.