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.
already solved
To solve this problem, we can use the formula for the nth term in an arithmetic sequence:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
Given that a_5 = 9 and a_14 = 18, we can substitute these values into the formula to form two equations:
a_5 = a_1 + (5-1)d
9 = a_1 + 4d .....(Equation 1)
a_14 = a_1 + (14-1)d
18 = a_1 + 13d .....(Equation 2)
Now, we can solve these two equations simultaneously to find the values of a_1 and d. Subtracting Equation 1 from Equation 2 gives:
18 - 9 = (a_1 + 13d) - (a_1 + 4d)
9 = 9d
Dividing both sides by 9, we find:
d = 1
Now, substitute the value of d = 1 into Equation 1 to solve for a_1:
9 = a_1 + 4(1)
9 = a_1 + 4
a_1 = 5
So, the first term, a_1, is 5 and the common difference, d, is 1.
Finally, we can find a_24 using the formula:
a_24 = a_1 + (24-1)d
a_24 = 5 + 23(1)
a_24 = 5 + 23
a_24 = 28
Therefore, the 24th term, a_24, is equal to 28.