The 3rd term in an arithmetic sequence is 12, the 7th term is 24,
a) How many common differences are there between a_3 and a_7?
b) What is the common difference of the sequence?
c) What is the first term in the sequence? a_1
d) What is the expression for this arithmetic sequence?
To answer these questions, we need to use the formula for an arithmetic sequence.
The formula for the nth term (a_n) in an arithmetic sequence is given by:
a_n = a_1 + (n - 1) * d
Where a_1 is the first term, n is the term number, and d is the common difference.
a) To find the number of common differences between a_3 and a_7, we subtract the term numbers:
7 - 3 = 4
Therefore, there are 4 common differences between a_3 and a_7.
b) To find the common difference, we can use any two terms in the sequence. Let's use a_3 and a_7:
a_3 = a_1 + (3 - 1) * d
a_7 = a_1 + (7 - 1) * d
We are given that a_3 = 12 and a_7 = 24. Plugging these values into the equations, we get:
12 = a_1 + 2d
24 = a_1 + 6d
Subtracting the first equation from the second equation, we can eliminate a_1:
24 - 12 = (a_1 + 6d) - (a_1 + 2d)
12 = 4d
d = 3
Therefore, the common difference of the sequence is 3.
c) To find the first term of the sequence (a_1), we can substitute the values of a_3 and d into the formula for the nth term:
a_3 = a_1 + (3 - 1) * d
Plugging in a_3 = 12 and d = 3, we have:
12 = a_1 + 2 * 3
12 = a_1 + 6
a_1 = 12 - 6
a_1 = 6
Therefore, the first term in the sequence (a_1) is 6.
d) Finally, to find the expression for this arithmetic sequence, we substitute the value of a_1 and d into the formula for the nth term:
a_n = a_1 + (n - 1) * d
Plugging in a_1 = 6 and d = 3, we get:
a_n = 6 + (n - 1) * 3
Therefore, the expression for this arithmetic sequence is a_n = 3n + 3.