(a display of a goods in a store has to be organized in layers so that the 7th layer , from the floor, consist of 28 items while the 18th layer must have 6 items.the number of items in the different layers forms an arithmetic sequence.) (1) give the expression, in terms of a and d, for the 7th term and 18th term?
To find the expression for the 7th term and 18th term in the arithmetic sequence, we need to determine the common difference (d) and the first term (a).
We know that the 7th layer has 28 items and the 18th layer has 6 items. We can use this information to set up two equations.
For the 7th term:
a + (7 - 1)d = 28
Simplifying this equation, we get:
a + 6d = 28 ---- (Equation 1)
Similarly, for the 18th term:
a + (18 - 1)d = 6
Simplifying this equation, we get:
a + 17d = 6 ---- (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (a and d). We can solve this system to find the expressions for a and d.
Subtracting Equation 2 from Equation 1, we eliminate a:
(a + 6d) - (a + 17d) = 28 - 6
Simplifying this equation, we get:
-11d = 22
Dividing both sides by -11, we find:
d = -2
Now that we have the value of d, we can substitute it back into either Equation 1 or Equation 2 to find the value of a.
Using Equation 1:
a + 6(-2) = 28
Simplifying this equation, we get:
a - 12 = 28
Adding 12 to both sides, we find:
a = 40
Therefore, the expression for the 7th term is:
a + (7 - 1)d = 40 + (7 - 1)(-2) = 26
And the expression for the 18th term is:
a + (18 - 1)d = 40 + (18 - 1)(-2) = -6