9) In an arithmetic sequence, each term is found by adding a common difference (d) to the previous term. We can use this fact to find a11.
From the given information, we know that a1 = -15 and a8 = 34. We can use the formula for the nth term in an arithmetic sequence to find d:
a8 = a1 + (n-1)d
34 = -15 + 7d
49 = 7d
d = 7
Now that we know d, we can use the formula for the nth term to find a11:
a11 = a1 + (n-1)d
a11 = -15 + 10d
a11 = -15 + 10(7)
a11 = 55
Therefore, a11 in the arithmetic sequence is 55.
10) To find the arithmetic mean (or average) of the sequence -10, ____, ____, ____, 18, we need to find the three missing terms.
We can use the fact that the sequence is arithmetic to find the common difference (d):
d = a2 - a1 = a3 - a2 = a4 - a3 = a5 - a4
We can see that the common difference is d = 28/3 or approximately 9.33.
To find the missing terms, we can add d to the previous term:
a2 = -10 + d = -10 + 28/3 = -2.67
a3 = a2 + d = -2.67 + 28/3 = 5.67
a4 = a3 + d = 5.67 + 28/3 = 13
a5 = a4 + d = 13 + 28/3 = 20.33
Now that we have all the terms, we can find the arithmetic mean:
mean = (a1 + a2 + a3 + a4 + a5)/5
mean = (-10 - 2.67 + 5.67 + 13 + 20.33)/5
mean = 5.4
Therefore, the arithmetic mean of the sequence is 5.4.