First, informally -- which can be more educational than doing the formal mathematics....
The 38th term is 15 terms after the 23rd term, and the value of the 38th term is 1 more than the value of the 23rd term. So the common difference in the sequence is 1/15.
The 41st term is 3 terms after the 38th term, so the value of the 41st term is 3 times 1/15 greater than the 38th term. 3*(1/15) = 3/15 = 1/5 = 0.2; the 41st term is 3+0.2 = 3.2.
ANSWER: 3.2
Formally....
if a is the first term and d is the common difference, then...
23rd term is a+22d
38th term is a+37d
a+37d=3
a+22d=2
Subtract the second equation from the first
15d=1
d=1/15
a+22d=2
a+(22/15)=2=30/15
a=8/15
The first term is 8/15; the common difference is 1/15.
The 41st term is a+40d = 8/15+40/15 = 48/15 = 16/5 = 3.2
ANSWER (again): 3.2
Note that, if we use the standard formal textbook method for solving the problem, we have to do a lot more work (to find the first term of the sequence) than we did using an informal method.
I'll discuss a slightly different approach to what greenestamps shows in his formal method.
d = common difference
23rd term is 2
24th term is 2+d
25th term is (2+d)+d = 2+2d
26th term is (2+2d)+d = 2+3d
...etc...
38th term is 2+(38-23)d = 2+15d
2+15d = 3
15d = 3-2
15d = 1
d = 1/15
41st term is
2+(41-23)d = 2+18d = 2+18*(1/15) = 16/5 = 3.2
Arithmetic sequences and linear functions are closely connected ideas.
x = term number = 1,2,3,...
y = the term itself
For instance the x = 23rd term is y = 2
(x,y) = (23,2) and (x,y) = (38,3) are two points on this line.
Use the slope formula to determine
m = (y2-y1)/(x2-x1)
m = (3-2)/(38-23)
m = 1/15
The slope is the common difference. This is because it tells us how to get from one term to the next (i.e. one point to the next). This is only when x is a positive whole number.
Then let's use point-slope form
y - y1 = m(x - x1)
y - 2 = (1/15)(x - 23)
y = (1/15)x - 23/15 + 2
y = (1/15)x - 23/15 + 30/15
y = (1/15)x + (-23 + 30)/15
y = (1/15)x + 7/15
y = (1/15)*(x+7)
To verify this equation works, try plugging in x = 23 and you should get y = 2.
If you plug in x = 38 then you should get y = 3. I'll let the student do these verifications.
If you plug in x = 41, then it leads to y = 16/5 = 3.2
In a number line, between the 38th term and 23rd term, there are 38-23 = 15 equal gaps,
so each gap is = .
Between 38th term and 41st tern, there are 3 gaps of the same length,
so the 41st term is
3 + = 3 + = 3 = 3.2. ANSWER