The following problem refers to an arithmetic sequence. If a_5=9 and a_14=18. Find the term a_1, the common difference and then find a_24.
a5 to a14 is 9 terms (differences)
... 18 - 9 = 9 ... so the common difference is ... 9/9 or 1
a5 is 4 terms (differences) from a1 ... a1 = a5 - 4 = 9 - 4 = 5
... you can check using a14
a24 is 23 differences from a1 ... you can calculate it
To solve this problem, we need to use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n - 1)d
First, let's find the common difference (d). Given that a_5=9 and a_14=18, we can substitute these values into the formula to get two equations:
9 = a_1 + (5 - 1)d ----(1)
18 = a_1 + (14 - 1)d ----(2)
Let's solve this system of equations:
From equation (1), we have:
a_1 + 4d = 9
From equation (2), we have:
a_1 + 13d = 18
Now, subtract equation (1) from equation (2) to eliminate a_1:
(a_1 + 13d) - (a_1 + 4d) = 18 - 9
a_1 + 13d - a_1 - 4d = 9
9d = 9
Divide both sides by 9:
d = 1
Now that we have found the common difference (d = 1), we can substitute it back into equation (1) to find the first term (a_1):
9 = a_1 + (5 - 1)(1)
9 = a_1 + 4
a_1 = 5
So, the first term of the arithmetic sequence is a_1 = 5, and the common difference is d = 1.
To find a_24, we can use the formula for the nth term again:
a_24 = a_1 + (24 - 1)d
a_24 = 5 + (23)(1)
a_24 = 5 + 23
a_24 = 28
Therefore, the 24th term of the arithmetic sequence is a_24 = 28.