To solve the problem, we need to use the formulas you provided and set up a system of equations.
Let's break down the problem and use the following variables:
- t1: the first term of the arithmetic sequence
- d: the common difference between the terms
We're given two pieces of information:
1. The sum of three consecutive terms is 165, which can be written as:
t1 + (t1 + d) + (t1 + 2d) = 165
2. The second term is 11 times the first term, which can be written as:
t1 + d = 11t1
Now, we can solve the system of equations:
From equation 2, we can rewrite it as:
d = 10t1
Substituting this value in equation 1, we get:
t1 + (t1 + 10t1) + (t1 + 20t1) = 165
4t1 + 30t1 = 165
34t1 = 165
t1 = 165/34
Now that we have the value of t1, we can find the common difference (d):
d = 10t1
d = 10 * (165/34)
Finally, we can find the three terms by plugging in the values of t1 and d:
First term: t1
Second term: t1 + d
Third term: t1 + 2d
Substituting the values:
First term: 165/34
Second term: 165/34 + 10 * (165/34)
Third term: 165/34 + 2 * 10 * (165/34)
Simplify these expressions, and you will have the three terms of the arithmetic sequence.