The sum of the second ans the sixth terms of an arithmetic sequence is 4.Thw third term is 24 more than the 11th term.

1Determine the first three terms in the sequence.

a+d + a+5d = 4

2a + 6d = 4
a+3d= 2

a+2d - (a+10d) = 24
a+2d-a-10d=24
-8d=24
d = -3

into a+3d = 2 ---> a - 9 = 2
a = 11

the first 3 terms are
11, 8, 5

..........................
check:
11 8 5 2 -1 -4 -7 -10 -13 - 16 - 19

2nd + sixth = 8 + (-4) = 4, that checks
third term = 5
11th term = -19 , difference = 24
all looks good

Let's denote the first term of the arithmetic sequence as "a" and the common difference as "d."

We know that the sum of the second and sixth terms is 4. This can be written as:

(a + 1d) + (a + 5d) = 4

Simplifying this equation, we have:

2a + 6d = 4 ...(equation 1)

We are also given that the third term is 24 more than the 11th term. This can be written as:

(a + 2d) = (a + 10d) + 24

Simplifying this equation, we have:

2d - 8d = 24

-6d = 24

Dividing both sides by -6, we get:

d = -4

Now, substituting the value of "d" into equation 1, we have:

2a + 6(-4) = 4

Simplifying, we get:

2a - 24 = 4

Adding 24 to both sides, we have:

2a = 28

Dividing both sides by 2, we get:

a = 14

Therefore, the first term (a) is 14 and the common difference (d) is -4.

To find the first three terms in the sequence, we can use the formula:

an = a + (n - 1)d

Substituting the values of a and d, we have:

a1 = 14
a2 = 14 + (2 - 1)(-4) = 10
a3 = 14 + (3 - 1)(-4) = 6

So, the first three terms in the sequence are 14, 10, and 6.

To determine the first three terms in the sequence, we need to use the given information.

Let's denote the first term of the arithmetic sequence as "a" and the common difference between terms as "d".

We are given that the sum of the second and sixth terms is 4. The formulas to find the terms of an arithmetic sequence are:

n-th term (Tn) = a + (n - 1)d
Sum of n terms (Sn) = (n/2)(2a + (n - 1)d)

Using these formulas, we can write the equations:
(T2) + (T6) = 4 ..................(1)
(T3) = (T11) + 24 ................(2)

Let's solve the equations step by step:

First, substitute the formulas for the 2nd and 6th terms into the first equation:
(a + d) + (a + 5d) = 4
2a + 6d = 4

Next, write the equation for the 3rd term using the formula:
(a + 2d) = (a + 10d) + 24
2a - 8d = 24

Now, we have a system of two equations with two variables. Solving this system will give us the values of "a" and "d", which will allow us to determine the first three terms of the sequence.

You can solve this system of equations using various methods such as substitution, elimination, or matrix operations. Once you find the values of "a" and "d", substitute them back into the formula for the n-th term (Tn) to find the first three terms.