An Arithmetic sequence has a tenth term of 17 and a fourteenth term of 30, ind the general term
PLEASE HELP!!!!
a+9d = 17
a+13d = 30
subtract them
4d = 13
d=13/4 = 3.25
then a + 9(3.25) = 17
a = -12.25
term(n) = a + (n-1)d
= -12.25 + (n-1)(3.25)
= -12.25 + 3.25n - 3.25
term(n) = 3.25n - 15.5
check:
term(10) = 32.5 - 15.5 = 17
term(14) = 14(3.25)-15.5 = 30 , all is good
thank you :)
To find the general term of an arithmetic sequence, you need to determine the common difference first.
The formula for the general term of an arithmetic sequence is:
tn = a + (n - 1)d
Where:
- tn is the nth term of the sequence
- a is the first term of the sequence
- d is the common difference
- n is the position of the term in the sequence
Given that the tenth term (tn) is 17 and the fourteenth term (tn) is 30, we can set up two equations based on the formula:
17 = a + 9d (equation 1)
30 = a + 13d (equation 2)
To solve these equations, we can use a method called "elimination." By subtracting equation 1 from equation 2, we can eliminate the "a" term:
30 - 17 = (a + 13d) - (a + 9d)
13 = 4d
Now, we have the value of the common difference (d).
Next, we substitute the value of d back into equation 1 or 2 to find the value of "a" (the first term). Let's use equation 1:
17 = a + 9(13/4)
17 = a + 9(3.25)
17 - 29.25 = a
-12.25 = a
Now that we have the values of "a" and "d," we can substitute them into the formula for the general term:
tn = a + (n - 1)d
tn = -12.25 + (n - 1)(13/4)
Simplifying further, we have:
tn = -12.25 + (13/4)n - 13/4
Finally, we can rewrite the equation in a more simplified form:
tn = (13/4)n - (49/4)
Therefore, the general term of the arithmetic sequence is tn = (13/4)n - (49/4).