I'm learning series and sequences (grade 11).

Please check that my steps show I understand what I'm doing/the concept and my answer as well:

5. The consecutive terms of an arithmetic sequences are 3.6, y, 8.2. Find the value of y.

This seems to be a sequence question but I used the formula for series.... Is there a better way to solve this question???

S3 = 3/2 (3.6+8.2)
=3/2(11.8)
=17.7

17.7 = 3/2(2 • 3.6 + (3-1)d)
17.7 = 3/2(7.2+2d)
17.7 = 10.8 + 3d
17.7 - 10.8 = 3d
6.9 = 3d
2.3 = d

So y = 5.9

I feel like I'm not doing this properly ... Please help! I have a quiz tomorrow on sequences ):

I also used the same method for this question:
Insert 3 evenly spaced numbers between -2 and 10. My final answer was 3 = d, so : -2, 1, 4, 7, 10.

What is the proper way to solve these questions? I don't think I'll be able to use series formulas on the quiz tomorrow which is only on sequences!

Also, I'm stuck on this question:

Find the 10th term of the arithmetic sequence where the first term is 5 and the 4th term is 17.

How would you solve this question?

Thank you so much in advance!

Consecutive numbers of an arithmetic have a common difference between them, that is ...

y - 3.6 = 8.2-y
2y = 11.8
y = 5.9

or
y must be "average" of the two other numbers
y = (3.6+8.2)/2 = 5.9

You sure went about it the long way.

For splacing 3 evenly spaced numbers between -2 and 10
-2, -, -, -, 10
then -2 must be the first term or a = -2
10 is th 5th term
term(5) = a + 4d
a+4d=10
-2+4d=10
4d=12
d=3
Again, your answer is correct

last question:
given: fifth term is 5 ---> a+4d = 5
4th term is 17 ----> a+3d = 17
subtract the two equations
d = -12
then in a+3d = 17
a + 3(-12) = 17
a= 17 + 36 = 53

term(10) = a+9d = 53 + 9(-12) = -55

check:
if a=53, d = -12, the first few terms are
53 41 29 17 5 ...
our answer is correct

Thank you so much Reiny!!

d doesnt = 12 because to subtract the 2 equations you would get 12d=18 then you whould have to devide and find that d=1.5

To find the value of y in the arithmetic sequence 3.6, y, 8.2, you can use the fact that in an arithmetic sequence, the difference between consecutive terms (denoted by d) is constant. Here's how you can solve it:

1. Start by finding the common difference (d) of the sequence:
d = 8.2 - 3.6 = 4.6

2. Now that you have the value of d, substitute it back into the sequence to find the missing term (y):
y = 3.6 + d = 3.6 + 4.6 = 8.2

So the value of y is 8.2.

For the question about inserting three evenly spaced numbers between -2 and 10, you can follow these steps:

1. Determine the common difference between consecutive terms:
d = (10 - (-2)) / (5 + 1) = 12 / 6 = 2

2. Starting with the first term (-2), add the common difference (d) to it successively to get the next three terms:
-2, -2 + 2 = 0, 0 + 2 = 2, 2 + 2 = 4, 4 + 2 = 6, 6 + 2 = 8, 8 + 2 = 10

So the sequence with three evenly spaced numbers between -2 and 10 is: -2, 0, 2, 4, 6, 8, 10.

Moving on to the question about finding the 10th term of an arithmetic sequence, given the first term (a_1) and the 4th term (a_4):

1. Start by finding the common difference (d) using the given terms:
a_4 = a_1 + 3d
17 = 5 + 3d

2. Solve the equation for d:
3d = 17 - 5
3d = 12
d = 4

3. Now, you can find the 10th term (a_10) using the arithmetic formula:
a_n = a_1 + (n-1)d
a_10 = 5 + (10-1)4
a_10 = 5 + 9*4
a_10 = 5 + 36
a_10 = 41

So the 10th term of the arithmetic sequence is 41.

Remember to practice more examples and work through the steps to solidify your understanding of sequences and series. Good luck with your quiz!