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.