the seventh term of a arithmetic sequence is 72 and the tenth term of the sequence is 90. if the sequence is defined by a formula f(n), then find f(2). How would you do this?
fn = a + d(n-1)
f(7) = a + 6 d = 72
f(10) = a + 9 d = 90
so
3 d = 90 - 72 = 18
d = 6
a + 6*6 = 72
a = 36
so our sequence is
f(n) = 36 + 6(n-1)
f(2) = 36 + 6(1)
f(2) = 42
To find the formula for the arithmetic sequence, we need to determine the common difference (d). The common difference is the constant value that is added to each term to obtain the next term in the sequence.
Given that the seventh term (f(7)) is 72 and the tenth term (f(10)) is 90, we can set up two equations using the formula for the nth term of an arithmetic sequence:
f(7) = a + 6d = 72, where a is the first term
f(10) = a + 9d = 90
By subtracting the first equation from the second equation, we can eliminate the 'a' term:
(a + 9d) - (a + 6d) = 90 - 72
3d = 18
d = 6
Now that we have the common difference (d), we can find the first term (a) by substituting it into either equation:
a + 6(6) = 72
a + 36 = 72
a = 36
So, the formula for the arithmetic sequence is f(n) = 36 + 6(n-1), where n represents the term number.
To find f(2), substitute n = 2 into the formula:
f(2) = 36 + 6(2-1)
f(2) = 36 + 6
f(2) = 42
Therefore, f(2) = 42.