Answer:
the 7th term of the arithmetic sequence is 10.
Step-by-step explanation:
To find an expression for the nth term (an) in an arithmetic sequence, we need to determine the common difference (d) between the terms.
We can start by using the given information:
a14 = 52 and a35 = 178.
Using the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1) * d,
we can create two equations using the given terms:
a14 = a1 + (14 - 1) * d = 52, (equation 1)
a35 = a1 + (35 - 1) * d = 178. (equation 2)
To solve this system of equations, we can subtract equation 1 from equation 2:
(35 - 1) * d - (14 - 1) * d = 178 - 52,
34d - 13d = 126,
21d = 126.
Dividing both sides of the equation by 21:
d = 6.
Now that we have the common difference (d), we can substitute it into equation 1 to find a1:
a1 = a14 - (14 - 1) * d,
a1 = 52 - 13 * 6,
a1 = 52 - 78,
a1 = -26.
Therefore, the expression for the nth term (an) of the arithmetic sequence is:
an = -26 + (n - 1) * 6.
To find the 7th term, we substitute n = 7 into the expression:
a7 = -26 + (7 - 1) * 6,
a7 = -26 + 6 * 6,
a7 = -26 + 36,
a7 = 10.
Therefore, the 7th term of the arithmetic sequence is 10.