find a6 for an arithmetic sequence where a1=3x+1 and d=2x+6.

2.A display of cans on a grocery shelf consists of 20 cans on the botton, 18 cans inthe next row, and so on in an arithmic sequence, until the to[p row has four cans. how many cans , in total, are in the display?

a = 3x+1

d = 2x+6

term(6) = a + 5d
= 3x+1 + 5(2x+6)
= 13x + 31

2.

a = 20
d = -2
term(n) = 4
4 = a + (n-1)d
4 =20 - 2(n-1)
4 = 20 - 2n + 2
2n = 18
n = 9

sum(9) = (9/2)(first + last)
= (9/2)(20+4) = 108

To find the value of a given term in an arithmetic sequence, we need to use the formula:

an = a1 + (n - 1)d

Where:
an = the nth term
a1 = the first term
n = the term number
d = the common difference

For the first question, we are given a1 = 3x + 1 and d = 2x + 6. We need to find a6.

a6 = a1 + (6 - 1)d
= (3x + 1) + (5)(2x + 6)
= 3x + 1 + 10x + 30
= 13x + 31

Therefore, a6 = 13x + 31.

For the second question, we need to find the total number of cans in the display.

We can see that the number of cans in each row form an arithmetic sequence. The first row has 20 cans, and the last row has 4 cans. We need to find the sum of this arithmetic sequence, which can be done using the formula for the sum of an arithmetic series:

Sn = (n/2)(a1 + an)

Where:
Sn = the sum of the first n terms
a1 = the first term
an = the nth term
n = the number of terms

In this case, the first term a1 = 20, the last term an = 4, and the common difference d = 18 - 20 = -2 (since the number of cans decreases by 2 each row).

We need to find the value of n, which represents the number of rows. We can use the formula:

an = a1 + (n - 1)d

4 = 20 + (n - 1)(-2)
4 = 20 - 2n + 2
2n = 18
n = 9

Thus, there are 9 rows of cans.

Now we can find the total number of cans in the display:

Sn = (n/2)(a1 + an)
= (9/2)(20 + 4)
= (9/2)(24)
= 108

Therefore, the total number of cans in the display is 108.