Answer:

The sequence given above is an arithmetic sequence.

First, we need to find the common difference d by using the equation

where is the first term and is the nth term.

Let n = 2, so = 7/3 or the second term and = 3

Substitute:

7/3 = 3 + (2-1) d

7/3 = 3 + 1d

(7/3) - 3 = (3 + 1d) -3 (Additive property of equality)

- 2/3 = d

Now that we know the common difference, we can now use the equation

to find the value of n, given that = -27

Substitue:

-27 = 3 + (n -1)(-2/3)

-27 = 3 - 2/3n + 2/3 (Distributive property)

2/3n = 3 + 2/3 + 27 (Additive property, here we add 2/3n and 27 on both sides)

2/3n = 92/3

n = 46 (Multiplicative inverse property in which we multiply both sides by 3/2)

Therefore, -27 is the 46th term of the arithmetic sequence.

a8 = 5

a21 = -60

n = 14 (number of terms from a8 to a21)

To get the common difference, use the nth term formula.

an = a1 + (n - 1)d

*Replace (an) to (a21) and (a1) to (a8)

a21 = a8 + (n - 1)d

-60 = 5 + (14 - 1)d

-60 = 5 + 13d

-60 - 5 = 13d

-65 = 13d

*divide both left and right side by 13 to get rid of the numerical coefficient of d

-5 = d

To get the 1st term, use the nth term formula

an = a1 + (n - 1)d

Given:

a8 = 5

d = -5

n = 8 (number of terms from a1 to a8)

*Replace (an) to (a8)

a8 = a1 + (n - 1)d

5 = a1 + (8 - 1)(-5)

5 = a1 + (7)(-5)

5 = a1 - 35

5 + 35 = a1

40 = a1

To get the 15th term, use nth formula again

Given:

a1 = 40

d = -5

n = 15 (number of terms from a1 to a15)

an = a1 + (n - 1)d

a15 = 40 + (15 - 1)(-5)

a15 = 40 + (14)(-5)

a15 = 40 - 70

a15 = -30

a27= a15+ (27-15)d

47= 29 + (12)d

47= 29 +12d

47-29= 12d

18= 12d

d= 3/2

a15= a5 +(15-5) 3/2

29= a5 + (10)3/2

29= a5 + 15

29-15= a5

a5= 14