Final answer:
To find the nᵗʰ term of an arithmetic sequence given the third and fifth terms, calculate the common difference and use the formula an = a1 + (n - 1)d. The nᵗʰ term is found to be an = -4 + (n - 1)(40).
Step-by-step explanation:
To find the nth term of the arithmetic sequence when given the third and fifth terms, we first determine the common difference d by subtracting the third term from the fifth term and then dividing by the number of terms between them.
The formula to find the nth term of an arithmetic sequence is an = a1 + (n - 1)d, where a1 is the first term.
We are given that the third term is 76 and the fifth term is 156. Thus:
a3 = a1 + 2d = 76
a5 = a1 + 4d = 156
Subtract the first equation from the second:
a5 - a3 = (a1 + 4d) - (a1 + 2d)
156 - 76 = 2d
d = 40
Now we can find the first term a1 using a3:
a1 = 76 - 2d
a1 = 76 - 2(40)
a1 = -4
Finally, we use the formula to find the nth term:
an = a1 + (n - 1)d
an = -4 + (n - 1)(40)